LIBRARY 

OF  THE 

UNIVERSITY  OF  CALIFORNIA. 
Class 


'  professional  3Ltfararg 

EDITED  BY  NICHOLAS  MURRAY  BUTLER 


THE  TEACHING 


OF 


ELEMENTARY   MATHEMATICS 


THE  TEACHING 


ELEMENTARY  MATHEMATICS 


BY 

DAVID   EUGENE   SMITH 

PROFESSOR   OF   MATHEMATICS   IN   TEACHERS'   COLLEGE,   COLUMBIA 
UNIVERSITY,   NEW   YORK. 


THE  MACMILLAN  COMPANY 

LONDON:  MACMILLAN  &  CO.,  LTD. 

1903 
All  rights  r€ttrvtd 


COPYRIGHT,  1900, 
Bv  THE  MACMILLAN  COMPANY. 


Set  up  and  electrotyped  February,  1900.     Reprinted  January, 
1901  ;  April,  1902;  September,  1903. 


J.  S.  Gushing  &  Co.  -  Berwick  &  Smith 
Norwood  Mass.  U.S.A. 


AUTHOR'S   PREFACE 

IT  is  evident  that  the  problem  of  preparing  a  work 
upon  the  teaching  of  elementary  mathematics  may  be 
attacked  from  any  one  of  various  standpoints.  A  writer 
may  confine  himself  to  model  lessons,  for  example;  or 
to  the  explanation  of  the  most  difficult  portions  of  the 
subject  matter  ;  or  to  the  psychology  of  the  subject ;  or 
to  the  comparison  of  historic  methods ;  or  to  the  exploit- 
ing of  some  hobby  which  he  has  ridden  with  success ;  or 
to  those  devices  which  occupy  so  much  time  in  the  ordi- 
nary training  of  teachers.  He  may  say,  and  with  truth, 
that  elementary  mathematics  now  includes  trigonom- 
etry, analytic  geometry,  and  the  calculus ;  and  that 
therefore  a  work  with  this  title  should  cover  the  ground 
of  Dauge's  "  Me"thodologie,"  or  of  Laisant's  masterly 
work,  "  La  Mathe*matique."  He  may  proceed  dogmati- 
cally, and  may  lay  down  hard  and  fast  rules  for  teaching, 
excusing  this  destruction  of  the  teacher's  independence 
by  the  thought  that  the  end  justifies  the  means.  But 
with  a  limited  amount  of  space  at  his  disposal,  what- 
ever point  of  attack  he  selects  he  must  leave  the 
others  more  or  less  untouched ;  he  cannot  condense 
an  encyclopedia  of  the  subject  in  three  hundred  pages. 


yi  AUTHOR'S  PREFACE 

Several  years  ago  the  author  set  about  to  find  some- 
thing of  what  the  world  had  done  in  the  way  of  making 
and  of  teaching  mathematics,  and  to  know  the  really 
valuable  literature  of  the  subject.  He  found,  however, 
no  manual  to  guide  his  reading,  and  so  the  accumulation 
of  a  library  upon  the  teaching  of  the  subject  was  a  slow 
and  often  discouraging  work.  This  little  handbook  is 
intended  to  help  those  who  care  to  take  a  shorter,  clearer 
route,  and  to  know  something  of  these  great  questions 
of  teaching,  —  Whence  came  this  subject  ?  Why  am  I 
teaching  it  ?  How  has  it  been  taught  ?  What  should 
I  read  to  prepare  for  my  work  ?  The  subject  is  thus 
considered  as  in  a  state  of  evolution,  while  comparative 
method  rather  than  dogmatic  statement  is  the  keynote. 
It  is  true  that  certain  types  are  suggested,  —  methods, 
they  are  often  called  ;  but  these  are  given  as  represent- 
ing the  present  development  of  the  subject,  and  not  as 
finalities.  The  effort  has  been,  throughout,  to  set  forth 
the  subject  as  in  a  state  of  progress  to  which  forward 
movement  the  teacher  is  to  contribute ;  we  have  quite 
enough  literature  representing  the  static  element. 

Considerable  attention  has  been  given  to  the  bibliog- 
raphy of  the  subject.  At  the  risk  of  being  accused  of 
going  beyond  the  needs  of  teachers,  the  author  has  sug- 
gested the  most  helpful  works  in  French  and  German, 
as  well  as  in  English,  and  has  not  hesitated  to  quote 
from  them.  The  body  of  the  page  is,  however,  always 
in  English,  —  the  footnotes  may  be  used  or  not,  as  the 


AUTHOR'S  PREFACE  vii 

reader  wishes.  Where  a  quotation  seemed  to  lose  some- 
thing by  being  put  into  English,  the  original  has  been 
placed  in  a  footnote.  By  these  references  the  reader  is 
put  in  touch  with  those  works  which  the  author  has 
found  of  great  value  to  him.  The  references  might 
easily  be  multiplied,  but  this  has  not  seemed  desirable. 
There  are  many  books  on  the  teaching  of  mathematics, 
some  of  them  quite  pretentious  in  their  claims,  a  few 
published  in  America,  a  few  in  England  and  France, 
and  a  large  number  in  Germany.  To  cite  all,  or  even 
a  majority  of  these,  might  be  positively  harmful ;  it  is 
hoped  that  the  selection  made  has  been  reasonably 
judicious. 

If  this  work  shall  help,  even  in  a  small  way,  to  open 
a  wider  field,  or  to  offer  a  better  point  of  view,  to  some- 
one just  entering  the  profession,  the  author  will  feel 
repaid  for  his  labors. 

DAVID  EUGENE  SMITH. 

STATE  NORMAL  SCHOOL,  BROCKPORT,  N.Y., 
January,  1900. 


EDITOR'S    INTRODUCTION 

PERHAPS  no  single  subject  of  elementary  instruction 
has  suffered  so  much  from  lack  of  scholarship  on  the 
part  of  those  who  teach  it  as  mathematics.  Arithmetic 
is  universally  taught  in  schools,  but  almost  invariably 
as  the  art  of  mechanical  computation  only.  The  true 
significance  and  the  symbolism  of  the  processes  em- 
ployed are  concealed  from  pupil  and  teacher  alike. 
This  is  the  inevitable  result  of  the  teacher's  lack  of 
mathematical  scholarship. 

The  subtlety,  delicacy,  and  accuracy  of  mathematical 
processes  have  the  highest  educational  value,  both 
direct  and  indirect.  To  treat  them  as  mechanical  rou- 
tine, not  susceptible  of  explanation  or  illumination  from 
a  higher  point  of  view,  is  to  destroy  in  large  measure 
the  value  of  mathematics  as  an  educational  instrument, 
and  to  aid  in  arresting  the  mental  development  of  the 
pupil. 

As  long  ago  as  the  time  of  Aristotle  it  was  pointed 
out  that  mathematics  should  not  be  defined  in  terms 
of  the  content  with  which  it  deals,  but  rather  in  terms 


X  EDITOR'S  INTRODUCTION 

of  its  method  and  degree  of  abstractness.  Kant  says  of 
mathematics,  in  the  "  Critique  of  Pure  Reason,"  "  The 
science  of  mathematics  presents  the  most  brilliant  ex- 
ample of  how  pure  reason  may  successfully  enlarge 
its  domain  without  the  aid  of  experience."1  He  then 
goes  on  to  point  out  the  ground  of  the  distinction 
between  philosophical  and  mathematical  knowledge, 
and  adds :  "  Those  who  thought  they  could  distinguish 
philosophy  from  mathematics  by  saying  that  the  former 
was  concerned  with  quality  only,  the  latter  with  quan- 
tity only,  mistook  effect  for  cause.  It  is  owing  to  the 
form  of  mathematical  knowledge  that  it  can  refer  to 
quanta  only,  because  it  is  only  the  concept  of  quantities 
that  admits  of  construction,  that  is,  of  a  priori  repre- 
sentation in  intuition,  while  qualities  cannot  be  repre- 
sented in  any  but  empirical  intuition."2 

Mr.  Charles  S.  Peirce  has  recently  made  the  criti- 
cism that  Kant  was  not  justified  in  supposing  that 
mathematical  and  philosophical  necessary  reasoning 
are  distinguished  by  the  circumstance  that  the  former 
uses  construction  or  diagrams.  Mr.  Peirce  holds  that 
all  necessary  reasoning  whatsoever  proceeds  by  con- 
structions, and  that  we  overlook  the  constructions  in 
philosophy  because  they  are  so  excessively  simple. 3 
He  goes  on  to  show  that  mathematics  studies  nothing 
but  pure  hypotheses,  and  that  it  is  the  only  science 

1  Miiller's  Translation  (New  York,  1896),  p.  572.        2  Ibid.,  p.  573. 
8  Educational  Review t  15,  214. 


EDITOR'S  INTRODUCTION  xi 

which  never  inquires  what  the  actual   facts  are.      It 
is  "the  science  which  draws  necessary  conclusions." 

This  acute  argument  is,  I  think,  at  fault  in  its  con- 
tention that  construction  is  employed  in  philosophical 
reasoning,  but  is  otherwise  sound.  It  fails,  however, 
to  point  out  clearly  these  facts :  — 

1.  The  human  mind  is  so  constructed  that  it  must 
see  every  perception  in  a  time-relation  —  in  an  order  — 
and  every  perception  of  an  object  in  a  space-relation 
—  as  outside  or  beside  our  perceiving  selves. 

2.  These   necessary  time-relations   are   reducible  to 
Number,  and  they  are  studied  in  the  theory  of  number, 
arithmetic  and  algebra. 

3.  These  necessary  space-relations  are  reducible  to 
Position  and  Form,  and  they  are  studied  in  geometry. 

Mathematics,  therefore,  studies  an  aspect  of  all 
knowing,  and  reveals  to  us  the  universe  as  it  presents 
itself,  in  one  form,  to  mind.  To  apprehend  this  and 
to  be  conversant  with  the  higher  developments  of 
mathematical  reasoning,  are  to  have  at  hand  the  means 
of  vitalizing  all  teaching  of  elementary  mathematics. 

In  the  present  book,  the  purpose  of  which  is  to 
present  in  simple  and  succinct  form  to  teachers  the 
results  of  mathematical  scholarship,  to  be  absorbed  by 
them  and  applied  in  their  class-room  teaching,  the 
author  has  wisely  combined  the  genetic  and  the  ana- 
lytic methods.  He  shows  how  the  elementary  mathe- 
matics has  developed  in  history,  how  it  has  been  used 


xii  EDITOR'S  INTRODUCTION 

in  education,  and  what  its  inner  nature  really  is.  It 
may  safely  be  asserted  that  the  elementary  mathe- 
matics will  take  on  a  new  reality  for  those  who  study 
this  book  and  apply  its  teachings. 

NICHOLAS  MURRAY   BUTLER. 

COLUMBIA  UNIVERSITY,  NEW  YORK, 
February  i,  1900. 


CONTENTS 


CHAPTER  I 

PAGE 

HISTORICAL  REASONS  FOR  TEACHING  ARITHMETIC.  —  Impor- 
tance of  the  question.  The  evolution  of  reasons.  The 
beginning  utilitarian.  Early  correlation.  Utilitarian  among 
trading  peoples.  Tradition  and  examinations.  The  cul- 
ture value.  As  a  remunerative  trade.  As  a  mere  show  of 
knowledge.  As  an  amusement.  As  a  quickener  of  the 
wit.  Scientific  investigation  of  reasons  .  .  ,  i-iS 


CHAPTER   II 

WHY  ARITHMETIC  is  TAUGHT  AT  PRESENT.  — Two  general 
reasons.  The  utility  of  arithmetic  overrated.  The  culture 
value.  Teachers  generally  fail  here.  Recognition  of  the 
culture  value.  What  chapters  bring  out  the  culture  value. 
What  may  well  be  omitted.  Relative  value  of  culture  and 
utility 19-41 


CHAPTER  III 

How  ARITHMETIC  HAS  DEVELOPED.  —  Reasons  for  studying 
the  subject.  Extent  of  the  subject.  The  first  step  — 
counting.  The  second  step  —  notation.  The  next  great 
step  in  arithmetic.  The  twofold  nature  of  ancient  arith- 
metic. Arithmetic  of  the  middle  ages.  The  period  of 
the  Renaissance.  Arithmetic  since  the  Renaissance.  The 
present  status  of  school  arithmetic  ....  42-70 
xiii 


xiv  CONTENTS 

CHAPTER   IV 

PAGE 

How  ARITHMETIC  HAS  BEEN  TAUGHT.  —  The  value  of  the 
investigation.  The  departure  from  object  teaching.  Rhym- 
ing arithmetics.  Form  instead  of  substance.  Instruction 
in  method.  Pestalozzi,  Tillich.  Reaction  against  Pesta- 
lozzi.  Grube.  Recent  writers  ....  71-97 

CHAPTER  V 

THE  PRESENT  TEACHING  OF  ARITHMETIC.  —  Objects  aimed 
at.  The  number  concept.  The  great  question  of  method. 
The  writing  of  numbers.  The  work  of  the  first  year. 
The  time  for  beginning  the  study.  Oral  arithmetic. 
Treating  the  processes  simultaneously.  The  spiral  method. 
Common  vs.  decimal  fractions.  Improvements  in  algorism. 
The  formal  solution.  Longitude  and  time.  Ratio  and 
proportion.  Square  root.  The  metric  system.  The  ap- 
plied problems.  Mensuration.  Text-books.  Explana- 
tions. Approximations.  Reviews  ....  98-144 

CHAPTER  VI 

THE  GROWTH  OF  ALGEBRA.  —  Egyptian  algebra.  Greek 
algebra.  Oriental  algebra.  Sixteenth  century  algebra. 
Growth  of  symbolism.  Number  systems.  Higher  equa- 
tions    145-160 

CHAPTER  VII 

ALGEBRA,  WHAT  AND  WHY  TAUGHT. — Algebra  defined.  The 
function.  Why  studied.  Training  in  logic.  Ethical 
value.  When  studied.  Arrangement  of  text-books  .  161-174 

CHAPTER  VIII 

TYPICAL  PARTS  OF  ALGEBRA.  —  Outline.  Definitions.  The 
awakening  of  interest.  Stating  a  problem.  Signs  of  aggre- 
gation. The  negative  number.  Checks.  Factoring.  The 


CONTENTS  XV 

PAGB 

remainder  theorem.  The  quadratic  equation.  Equiva- 
lent equations.  Extraneous  roots.  Simultaneous  equa- 
tions and  graphs.  Methods  of  elimination.  Complex 
numbers.  The  applied  problems.  The  interpretation  of 
solutions 175-223 

CHAPTER   IX 

THE  GROWTH  OF  GEOMETRY.  —  Its  historical  position.  The 
dawn  of  geometry.  Geometry  in  Egypt ;  in  Greece.  Re- 
cent geometry.  Non-Euclidean  geometry  .  .  224-233 

CHAPTER  X 

WHAT  is  GEOMETRY?  GENERAL  SUGGESTIONS  FOR  TEACH- 
ING. —  Geometry  defined.  Limits  of  plane  geometry. 
The  reasons  for  studying.  Geometry  in  the  lower  grades. 
Intermediate  grades.  Demonstrative  geometry.  The  use 
of  text-books 234-256 

CHAPTER  XI 

THE  BASES  OF  GEOMETRY.  —  The  bases.    The  definitions. 

Axioms  and  postulates 257-270 

CHAPTER  XII 

TYPICAL  PARTS  OF  GEOMETRY.  —  The  introduction  to  demon- 
strative geometry.  Symbols.  Reciprocal  theorems.  Con- 
verse theorems.  Generalization  of  figures.  Loci  of  points. 
Methods  of  attack.  Ratio  and  proportion.  The  impos- 
sible in  geometry.  Solid  geometry  ....  271-296 

CHAPTER  XIII 

THE  TEACHER'S  BOOK-SHELF.  —  Arithmetic.  Algebra.  Geom- 
etry. History  and  general  method  ....  297-305 


INDEX 307-312 


THE 

TEACHING   OF   ELEMENTARY 
MATHEMATICS 

CHAPTER   I 
HISTORICAL  REASONS  FOR  TEACHING  ARITHMETIC 

Importance  of  the  question  —  For  one  who  is  pre- 
paring to  teach  any  particular  branch,  and  who  hopes 
for  success,  the  most  important  question  is  this :  Why 
is  the  subject  taught  ?  More  important  than  all  meth- 
ods, more  important  than  all  devices  or  questions 
of  text-books,  or  advice  of  the  masters,  is  this  far* 
reaching  inquiry.  Upon  the  answer  depends  the  solu- 
tion of  the  problems  relating  to  the  presentation  of  the 
subject,  the  grade  in  which  it  should  be  begun,  the 
time  it  should  consume,  the  text-books,  the  methods, 
the  devices,  —  in  fine,  the  general  treatment  of  the 
whole  matter  in  hand.  It  is  the  old,  old  cry,  "We 
know  not  whither  Thou  goest,  and  how  can  we  know 
the  way  ? "  Unless  the  goal  is  known,  what  hope 
has  one  to  find  the  path? 

Of  course  the  inquiry  is  of  no  interest  to  the  ma- 
chine teacher,  the  teacher  who  is  content  to  follow 

B  I 


2       THE  TEACHING   OF   ELEMENTARY  MATHEMATICS 

the  book  unthinkingly,  to  see  the  old  curriculum  re- 
main forever  unchanged,  and  to  follow  the  path  his 
teacher  trod,  even  though  it  be  rough  to  the  foot  and 
without  interest  to  the  eye.  But  in  England  and 
America  to-day  we  have  a  host  of  young  and  enthu- 
siastic teachers  who  are  anxious  to  make  the  Anglo- 
Saxon  educational  system  the  best,  and  who  are 
willing  to  inquire  and  to  experiment.  For  such 
teachers  this  question  is  vital. 

The  evolution  of  reasons  —  This  search  after  reasons 
may  be  pursued  either  from  the  standpoint  of  a  mere 
inquirer  into  the  conditions  of  to-day,  or  from  that  of 
one  who  is  interested  in  the  evolution  of  the  ideas 
which  are  now  in  favor.  While  it  is  not  possible  in 
a  work  of  this  nature  to  enter  into  the  details  of  the  de- 
velopment of  the  reason  for  the  presence  of  arithmetic 
in  the  curriculum  to-day,  some  slight  reference  to  this 
development  may  be  of  interest,  and  should  be  of  value. 

The  beginning  utilitarian  —  In  the  far  East,  and 
in  the  far  past,  the  reason  for  teaching  arithmetic  to 
children  was  almost  always  purely  utilitarian.  To  the 
philosopher  it  was  more  than  this,  but  in  the  early 
Chinese  curricula  it  was  given  place  merely  that  the 
boy  might  have  sufficient  knowledge  of  the  four  fun- 
damental processes  for  the  common  vocations  of  life.1 

1  Schmid,  K.  A.,  Geschichte  der  Erziehung  vom  Anfang  an  bis  auf 
unsere  Zeit,  Stuttgart,  1884-98,  Vol.  I,  p.  78.  Hereafter  referred  to  as 
Schmid. 


HISTORICAL  REASONS  FOR  TEACHING  ARITHMETIC      3 

This  was  done  in  the  common  schools  almost  from 
the  first,  but  in  the  middle  ages1  the  subject  so  in- 
creased in  importance  that  special  schools  were  estab- 
lished for  the  study  of  arithmetic.  A  little  later2  it 
was  taught  as  a  special  course  in  the  high  schools, 
open  to  those  who  had  a  taste  in  this  direction, 
although  even  then  children  must  have  continued  to 
learn  common  reckoning  in  the  earlier  years.  In 
general,  however,  it  has  been  taught  in  the  far  East 
for  two  thousand  years,  because  of  the  utilities  which 
it  possesses,  or  merely  for  the  purposes  of  examina- 
tion, or  because  it  correlated  with  a  study  of  the 
sacred  books.3 

Early  correlation  —  In  India  little  could  be  expected 
for  arithmetic  in  the  schools.  The  aim  of  education, 
as  summarized  in  the  first  book  of  Manu,  was  to  bring 
man  to  lead  a  religious  life.  The  reading  of  the  Veda, 
the  giving  of  alms,  these  were  fundamental  features  of 
education.4  Even  to-day  is  this  the  case.  For  more 
than  two  thousand  years  the  curriculum  and  the 
methods  have  remained  quite  unchanged,  and  even 
in  our  day,  in  the  native  schools,  the  boy's  work  is 
largely  that  of  memorizing  the  Hindu  scriptures  and 

1  Under  the  Sung  dynasty,  961-1280.     Schmid,  I,  p.  80. 
*  Under  the  Ming  dynasty,  1368-1644. 

8  Laurie,  S.  S.,  Historical  Survey  of  Pre-Christian  Education,  London, 
1895,  p.  128,  141,  148.     Hereafter  referred  to  as  Laurie. 
4  Schmid,  I,  p.  105-107. 


4       THE  TEACHING  OF  ELEMENTARY   MATHEMATICS 

picking  up  other  knowledge  incidentally,  a  classical 
example  of  extreme  correlation.  For  such  people, 
arithmetic,  beyond  the  mere  rudiments,  is  of  value 
only  as  it  throws  light  upon  the  central  subject,  and 
hence  it  has  little  place  in  the  curriculum.1 

The  same  idea  characterized  the  early  Mohammedan 
schools,  where  the  Koran  furnished  the  core  of  instruc- 
tion, a  plan  of  education  still  obtaining,  on  a  slightly 
more  liberal  scale,  in  the  present  schools  of  Islam.2  It 
also  held  quite  general  sway  in  the  monastic  schools 
of  the  middle  ages,  where  arithmetic,  like  everything 
else,  was  either  warped  to  correlate  with  theology,  or 
confined  to  the  simplest  calculations.3  That  arithmetic 
was  popularly  considered  merely  as  having  some  slight 
value  in  trade  is  shown  by  a  familiar  bit  of  monkish 
doggerel,  as  old  at  least  as  the  beginning  of  the  fifteenth 
century.4  It  thus  sets  forth  the  values  of  the  seven 
liberal  arts,  —  grammar,  dialectic,  rhetoric,  music,  arith- 
metic, geometry,  and  astronomy : 

"  Gramm.  loquitur,  Dia.  vera  docet,  Rhe.  verba  colorat ; 
Mus.  canit,  Ar.  numerat,  Ge.  ponderat,  As.  colit  astra." 

1  For  a  description  of  the  arithmetic  in  the  native  Hindu  schools  of  the 
present  consult  Delbos,  L.,  Les  Mathematiques  aux  Indes  Orientales,  Paris, 
1892,  —  pamphlet. 

2  Schmid,  II  (i),  p.  599. 

8  Ib.,  II  (i),  p.  86.  In  this  line  is  the  rule  attributed  to  Pachomius, 
"  Omnino  nullus  erit  in  monasterio,  qui  non  discat  literas  et  de  scripturis 
aliquid  teneat." 

*Ib.,  II  (i),  p.  114. 


HISTORICAL   REASONS  FOR  TEACHING  ARITHMETIC      5 

For  the  mediaeval  cloister  schools  the  computation 
of  Easter  day  was  the  one  great  problem.  On  this 
depended  the  other  movable  feasts,  and  every  monastery 
was  under  the  necessity  of  having  someone  who  knew 
enough  of  calculating  to  determine  this  date.1 

Utilitarian  among  trading  peoples  —  Among  the 
Semitic  peoples  we  find  arithmetic  more  extensively 
taught.  The  Semite  has  generally  interested  himself 
not  in  the  thing  for  its  own  sake,  but  for  what  it 
contained  for  him  in  a  practical  way.  Hence  the 
Assyrians  and  Arabs  and  related  peoples  have  no 
national  epos  and  no  enduring  art.2  But  they  found 
in  arithmetic  a  subject  usable  in  trade,  and  hence  it 
was  extensively  taught  in  their  schools.  Among  the 
ruins  in  and  about  ancient  Babylon  it  is  not  uncom- 
mon to  find  tablets  containing  extensive  bank  ac- 
counts, and  lately  some  interesting  specimens  of 
pupils'  work  in  arithmetic  have  come  to  light.8 

Among  the  Jews,  after  elementary  instruction  was 
made  obligatory,4  arithmetic  formed,  with  writing  and 
the  study  of  the  Pentateuch,  the  sole  work  from  the 
sixth  to  the  tenth  year  of  the  child's  school  life. 

1  Rashdall,  H.,  Universities  of  Europe  in  the  Middle  Ages,  I,  p.  35. 
Schmid,  II  (i),  p.  117. 

a  Schmid,  I,  p.  142. 

8  Ib.,  I,  p.  152,  153.  The  firm  of  Egibi  and  Sons  is  often  mentioned  in 
these  tablets;  it  was  long  famous  in  banking  business  from  Nebuchadnez- 
zar's time  on. 

4  A.D.  64.     Laurie,  p.  97. 


6       THE  TEACHING  OF  ELEMENTARY   MATHEMATICS 

Even  in  Greece,  and  among  the  philosophers, 
where  one  would  expect  something  beyond  the  mere 
necessities  of  existence,  arithmetic  was  not  in  general 
highly  valued.  Socrates,  who  recommends  the  sub- 
ject in  the  curriculum,  does  so  with  a  warning  against 
carrying  it  beyond  the  needs  of  common  life.  Of 
course  among  the  Spartans,  who  trained  for  war,  the 
science  had  no  place.1 

In  Rome,  a  city  of  commerce  and  of  war,  the  sub- 
ject was  naturally  looked  upon  as  of  merely  utilita- 
rian importance.  The  vast  commercial  interests  of 
the  city,  extending  to  the  farthest  corner  of  the 
great  empire,  made  a  business  education  imperative 
for  a  large  class.  Arithmetic  nourished,  but  merely 
as  the  drudgery  of  calculation.  So  Cicero  tells  us 
that  in  his  time  the  Romans  esteemed  only  practical 
reckoning,  nor  was  the  learned  Boethius,  the  philo- 
sopher, ecclesiastic,  and  mathematician,  able  to  raise 
it  to  any  higher  plane.2 

In  the  cloisters,  when  not  taught  for  the  purposes 

1  Girard,  Paul,  L'Education  Athenienne  au  Ve  et  au  IV6  siecle  avant 
J.  C,  2.   ed.,  Paris,   1891,  p.   136-138;     Martin,   Alex.,    Les  Doctrines 
Pedagogiques  des  Grecs,  Paris,  1881,  p.  12;   Schmid,  I,  p.  231,  232. 

2  Laurie,  p.  360  ;  Clarke,  G.,  The  Education  of  Children  at  Rome,  New 
York,  1896,  p.  1 6,  17,  85  ;    Sterner,   M.,   Geschichte   der  Rechenkunst, 
Miinchen,  1891,  p.  73,  hereafter   referred  to  as  Sterner ;    Schmidt,  K., 
Geschichte  der  Padagogik,  Cothen,  1873,  I,  p.  408  ;  Dittes,  F.,  Geschichte 
der  Erziehung  und  des  Unterrichts,  9.  AufL,  Leipzig,  1890,  p.  73  ;  Schmid, 
II  (i),  p.  140. 


HISTORICAL  REASONS   FOR  TEACHING  ARITHMETIC      7 

of  computing  Easter  or  as  a  "whetstone  of  wit," 
arithmetic  was  considered  as  merely  of  value  in 
trade.  Even  Beda,  one  of  the  best  teachers  of  his 
time,  looked  upon  the  subject  as  purely  utilitarian.1 
During  the  middle  ages,  too,  there  was  a  great 
revival  of  trade  and  a  corresponding  revival  of  com- 
mercial arithmetic.  For  a  long  time  after  the  close 
of  the  thirteenth  century  Northern  Italy  was  the 
gateway  for  trade  entering  Europe  from  the  Orient. 
Thence  it  passed  northward,  through  Augsburg, 
Niirnberg,  and  Frankfurt  am  Main,  to  Leipzig  and 
the  northern  Hanseatic  towns  on  the  east,  and  to 
Cologne  and  the  Netherlands  on  the  west.  Similarly 
in  France,  Lyons  and  Paris,  and  in  Austria,  Vienna, 
Linz,  and  Ofen,  became  important  commercial  cen- 
tres. But  Italy  was  par  excellence  the  mercantile 
nation  and  the  source  of  commercial  arithmetic,  and 
we  find  the  utilitarian  influence  supreme,  from  the 
source  all  along  this  pathway  of  commerce.2  It  was 
among  the  merchants  along  this  path  of  trade  that 
as  early  as  the  thirteenth  century  a  feeling  of  dis- 
satisfaction arose  against  the  arithmetical  training 
of  the  Church  schools.  Mysticism  and  formalism 
had  so  supplanted  religion,  to  say  nothing  of  other 

1  Schmid,  II  (i),  p.  140. 

2  Unger,  F.,  Die  Methodik  der  praktischen  Arithmetik  in  historischer 
Entwickelung  vom  Ausgange  des  Mittelalters  bis  auf  die  Gegenwart, 
Leipzig,  1888,  p.  3  seq.     Hereafter  referred  to  as  Unger. 


8       THE  TEACHING  OF  ELEMENTARY   MATHEMATICS 

subjects  of  study,  that  even  the  common  people  were 
wont  to  point  with  shame  to  the  results  of  monastic 
training.1  Even  when  the  universities  began  to 
spring  up,  about  noo,2  and  arithmetic  might  hope  to 
break  away  from  the  bonds  of  commerce,  there  was 
little  improvement.  Scholasticism,  disputations,  philo- 
sophic hair-splitting  —  these  had  little  use  for  a  sub- 
ject like  this.  One  who  had  made  a  little  progress 
in  fractions  was  a  mathematician.  Save  as  leading 
to  the  calculations  of  the  calendar,  and  as  it  might 
occasionally  touch  the  Aristotelian  philosophy,  mathe- 
matics had  no  standing.3 

It  was  during  this  mediaeval  period   that  the  Han- 
seatic  league  became  a  power.     This  great  trust  —  for 

1  Schmid,  II  (i),  p.  312. 

2  Laurie,  S.  S.,  The  rise  of  Universities,  lect.  vi. 
8  "  Omnis  hie  excluditur,  omnis  est  abiectus, 

Qui  non  Aristotelis  venit  armis  tectus." 

Chartular.  Univ.  Paris,  I,  Introd.,  p.  xviii. 

Schmid,  II  (l),  p.  427,  447,  448.  In  Cologne  in  1447  the  outlook  for 
mathematics,  as  indeed  for  other  subjects,  was  exceedingly  poor  if  one 
may  judge  from  the  verses  in  Horatian  measure  of  the  young  Conrad 

Celtes: 

"  Nemo  hie  latinam  grammaticam  docet, 

Nee  explotis  rhetoribus  studet, 

Mathesis  ignota  est,  figuris 

Quidque  sacris  numeris  recludit. 

Nemo  hie  per  axem  Candida  sidera 

Inquirit,  aut  quae  cardinibus  vagis 

Moventur,  aut  quid  doctus  alta 

Contineat  Ptolemaeus  arte."  —  Schmid,  II  (i),  p.  449. 


HISTORICAL   REASONS   FOR  TEACHING  ARITHMETIC     9 

such  it  may  be  styled  —  soon  found  that  it  was  neces- 
sary to  establish  its  own  schools  if  it  wished  a  prac- 
tical education  for  the  rising  generations.  And  so 
there  was  to  be  found  in  each  town  of  any  size  along 
the  highway  dominated  by  the  league,  an  arithmetic 
master  (Rechenmeister),  who  held  the  monopoly  of 
teaching  the  subject  there.  Not  unfrequently  was 
the  Rechenmeister  also  the  city  accountant,  treasurer, 
sealer  of  weights  and  measures,  etc.  It  was  natural, 
therefore,  that  arithmetic  should  tend  to  become  a 
purely  utilitarian  subject  in  these  places,  and  so  in 
great  measure  it  was.  It  is  interesting  to  recall  that 
the  last  of  the  Rechenmeisters,  Zacharias  Schmidt  of 
Nurnberg,  kept  his  place  until  I82I.1  As  late  as  the 
sixteenth  century,  when  the  reformers  began  to  do 
some  thinking  in  education,  in  a  school  as  famous  as 
the  Strassburg  gymnasium,  Johann  Sturm,  in  his  cur- 
riculum of  1565,  makes  no  mention  of  arithmetic  in 
his  entire  ten  years'  course,  so  completely  commer- 
cial had  the  subject  become.2 

To  refer  more  specifically  to  the  universities,  even 
at  Cambridge,  which  already  in  the  middle  ages  led 
Oxford  in  mathematical  teaching,  arithmetic  had 
scarcely  any  attention.8  At  Oxford  during  this  period 

1  Unger,  p.  26,  33. 

2  Paros,  Jules,  Histoire  universelle  de  la  Pedagogic,  p.  126;    Schmid, 
II  (2),  p.  325. 

•  Rashdall,  H.,  Universities  of  Europe  in  the  Middle  Ages,  II,  p.  556. 


IO     THE  TEACHING  OF  ELEMENTARY  MATHEMATICS 

a  term  in  Boethius  was  all  that  was  required.1  Even 
when  a  chair  of  arithmetic  was  founded  in  the  Uni- 
versity of  Bologna,  a  school  which  owed  its  promi- 
nence in  mathematics  to  Arabo-Greek  influence,  it 
was  little  more  than  that  of  a  surveyor  and  general 
computer.2  In  Paris  the  subject  had  no  hold,  and  in 
Vienna,  where  more  was  done  than  in  the  Sorbonne, 
only  a  nominal  amount  of  arithmetic  was  required.3 
In  general,  mathematics  was  looked  upon  as  a  light 
subject  in  the  mediaeval  universities. 

Tradition  and  examinations  —  The  Egyptian  reason 
for  teaching  arithmetic  may  be  seen  in  the  interesting 
account  of  a  school  of  the  fourteenth  century  B.C., 
given  by  the  late  Dr.  Ebers  in  the  second  chapter  of 
Uarda.4  Here,  where  the  'life  and  thought  of  the 
people,  so  closely  joined  to  the  river  with  its  periodic 
mystery  of  rise  and  fall,  naturally  took  on  regularity, 
rule,  canonical  form,  and  mysticism,  educational  prog- 
ress could  only  come  from  renewed  intercourse  with 
the  outer  world.  Hence  arithmetic  came  to  be  taught 
merely  as  a  matter  of  custom,  of  tradition  as  fixed  as 
human  law  can  be.  It  was  required  for  examinations, 
and  the  examiner  followed  a  certain  line;  hence,  the 


1  Rashdall,  H.,  Universities  of  Europe  in  the  Middle  Ages,  II,  p.  457. 
2Ib.,  II,  p.   243,  66 1  n.;   I,  p.  249. 

8  For  the  B.  A.  degree,  "  Primum  librum  Euclidis  .  .  .  aliquem  librum 
in  arithmetica."     Ib.,  II,  p.  240,  674. 
*  See  also  Schmid,  I,  p.  172. 


HISTORICAL  REASONS  FOR  TEACHING  ARITHMETIC     1 1 

student  must  be  prepared  along  that  line.1  This  is 
always  the  tendency  under  a  centralized  examina- 
tion system,  or  where  an  inflexible  official  programme 
must  be  followed.  As  M.  Laisant  says,  "a  pro- 
gramme is  always  bad,  essentially  because  it  is  a 
programme." 

An  excellent  illustration  of  the  petrifying  tendency 
of  such  an  examination  system  has  recently  come  to 
light.  The  oldest  deciphered  work  on  mathematics 
is  a  papyrus  manuscript  preserved  in  the  British 
Museum.  It  was  copied  by  one  Ahmes  (Aahmesu, 
the  Moonborn),  a  scribe  of  the  Hyksos  dynasty,  say 
between  2000  and  1700  B.C.,  from  an  older  work  dat- 
ing from  2400  B.C.2  Without  going  into  details  as 
to  the  contents  of  the  work,  it  answers  the  present 
purposes  to  say  that  the  arithmetical  part  was  de- 
voted chiefly  to  unit  fractions.  Instead  of  writing  the 
fraction  ^  (using  modern  notation)  Ahmes  and  his 
predecessor  write  it  -^  +  -fa  +  y^T-  Now,  within  the 
past  decade  there  have  been  found  in  Kahun,  near 

»  Schmid,  I,  p.  173  ;  Laurie,  p.  44. 

a  That  is,  from  the  reign  of  Araenemhat  III,  2425-2383  B.C.  Cantor, 
M.,  Vorlesungen  fiber  Geschichte  der  Mathematik,  I,  p.  21,  n.  This  work, 
the  standard  authority  in  the  history  of  mathematics,  will  hereafter  be 
referred  to  as  Cantor;  Vol.  I,  2.  Auf.,  1894,  Vol.  II,  1892,  Vol.  Ill,  1898, 
Leipzig.  The  Ahmes  papyrus  was  translated  and  published  by  Eisenlohr, 
A.,  Ein  mathematisches  Handbuch  der  alten  Aegypter,  Leipzig,  1877, 
and  an  English  edition  has  recently  appeared.  A  brief  summary  is  given 
in  Gow,  J.,  A  short  History  of  Greek  Mathematics,  Cambridge,  1884,  p.  15, 
hereafter  referred  to  as  Gow, 


12     THE  TEACHING  OF  ELEMENTARY  MATHEMATICS 

the  pyramids  of  Illahum,  two  mathematical  papyri 
treating  fractions  exactly  after  the  manner  of  Ahmes, 
and  there  has  been  published  in  Paris  an  interesting 
papyrus  found  in  the  necropolis  of  Akhmim,  the 
ancient  Panopolis,  in  Upper  Egypt,  written  by  a 
Christian  Greek  somewhere  from  the  fifth  to  the  ninth 
century  A.D.  In  this  latter  work,  also,  fractions  are 
treated  just  as  Ahmes  had  handled  them  over  two 
thousand  years  before.1  The  illustration  is  extreme, 
but  it  shows  the  tendency  of  tradition,  of  canonical 
laws,  and  of  the  examination  system,  which  for  so 
many  centuries  dominated  the  civil  service  of  Egypt. 
The  culture  value  —  Occasionally,  however,  even  in 
ancient  times,  there  appeared  a  suggestion  of  a  higher 
reason  for  the  study  of  arithmetic.  Solon  and  Plato 
saw  in  the  subject  an  opportunity  for  training  the 
mind  to  close  thinking,  the  former  placing  here  its 
greatest  value,  and  the  latter  asserting  that  even  the 
most  elementary  operations  contributed  to  the  awaken- 
ing of  the  soul  and  to  stirring  up  "a  sleepy  and  un- 
instructed  spirit.  We  see  from  the  Platonic  dialogues 
how  mathematical  problems  employed  the  mind  and 
thoughts  of  young  Athenians."2  Plato  even  goes  so 


1  Baillet,  J.,   Le  papyrus  mathematique  d' Akhmim,  Paris,  1892,  in  the 
Me"moires  .  .  .  de  la  mission  archeologique  fran9aise  au  Caire. 

2  Browning,  Oscar,  Educational  Theories,  New  York,  1882,  p.  6;  Mar- 
tin, Alexandre,  Les  Doctrines  Pedagogiques  des  Grecs,  Paris,  1881,  p.  44; 
Schmid,  I.  p.  233. 


HISTORICAL   REASONS  FOR  TEACHING  ARITHMETIC     13 

far  as  to  wish  arithmetic  taught  to  girls,  and  Aristotle 
also  champions  the  higher  cause  when  he  asserts  that 
"children  are  capable  of  understanding  mathematics 
when  they  are  not  able  to  understand  philosophy." 
Still,  in  Aristotle's  scheme  of  state  education  we  look 
in  vain  for  any  details  as  to  the  carrying  out  of  the 
idea  here  expressed.1  Naturally,  too,  Pythagoras,  the 
first  great  mathematical  master,  saw  in  arithmetic 
something  beyond  mere  calculation.  "Gymnastics, 
music,  mathematics,  these  were  the  three  grades  of  his 
educational  curriculum.  By  the  first  the  pupil  was 
strengthened;  by  the  second  purified;  by  the  third 
perfected  and  made  ready  for  the  society  of  the 
gods."2 

In  the  middle  ages  the  same  feeling  occasionally 
crops  out,  as  when  ^Eneas  Sylvius  (later  Pope  Pius 
II,  from  1458  to  1464),  the  apostle  of  humanism  in 
Germany,  advocated  the  study  of  arithmetic  for  its 
own  sake,  provided  it  should  not  require  too  much 
time.  Humanism  failed,  however,  to  advance  math- 
ematics to  any  great  extent  in  the  learned  schools. 
With  few  exceptions  this  task  was  left  to  the  tech- 
nical schools.  Occasionally  some  leader  like  Stehn 
was  far-sighted  enough  to  appreciate  in  a  slight 

1  Davidson,  Thomas,  Aristotle  and  Ancient  Educational  Ideals,  New 
York,  1 892,  p.  198. 

2  Ib.,  p.  100.     But  see  Mahaffy,  P.  J.,  Old  Greek  Education,  New 
York,  1882,  p.  89,  on  the  slight  influence  of  Pythagoras  on  education. 


14     THE  TEACHING  OF  ELEMENTARY   MATHEMATICS 

degree  the  educational  value  of  the  subject,  but  such 
cases  were  rare.1 

As  a  remunerative  trade  —  In  the  development  of 
the  science  there  have  been  periods  in  which  it  was 
not  uncommon  for  mere  problem-solvers  to  undertake 
arithmetical  puzzles  for  pay,  and  occasionally  arith- 
metic has  been  studied  with  this  in  view,  although  of 
course  to  no  great  extent.  Hans  Conrad,  a  friend  of 
Adam  Riese  the  famous  German  arithmetician  (1492- 
1559),  solved  problems  for  pay.  Also  in  the  time  of 
the  early  Italian  algebraists,  Scipione  del  Ferro,  An- 
tonio del  Fiore,  Tartaglia,  and  Cardan,  the  same  state 
of  affairs  existed ;  it  was  a  period  of  secret  rules,  and 
learning  was  neither  open  nor  free.2 

As  a  mere  show  of  knowledge  —  This  has  not  unf  re- 
quently  been  one  of  the  most  apparent  of  reasons,  and 
especially  so  in  the  Latin  schools  of  the  sixteenth  cen- 
tury. Thus  Gemma  Frisius,  one  of  the  most  famous 
text-book  writers  of  his  time,  presents  as  the  second 
number  in  his  arithmetic,  23456345678,  "vicies  &  ter 
millies  millena  millia,  quadringenta  quinquaginta  sex 
millena  millia,  trecenta  quadraginta  quinque  millia, 
sexcenta  &  septuaginta  octo."3  Such  a  display  of  words 

1  Stehn  (Johannes  Stenius)  writes,  in  Wittenberg  in  1594,  "Num  dis- 
ciplina  numerorum  Methodica  iure  possit  exulare  Scholis  puram  et  solidam 
Philosophiam  ambientibus. "  Schmid,  II  (2),  p.  373. 

2  Unger,  p.  33,  34. 

8  Arithmeticae  Practicae  Methodus  Facilis,  edn.  of  1551,  p.  A.  v. 


HISTORICAL  REASONS  FOR  TEACHING  ARITHMETIC     15 

cannot  be  dignified  by  the  term  knowledge ;  it  is  only 
a  pretence.  It  has  its  counterpart  in  the  absurdly  ex- 
tended number  names  in  some  of  our  present  arith- 
metics and  in  subjects  like  compound  proportion. 

As  an  amusement  —  Arithmetic  has  also  been  taught 
for  its  amenities,  and  in  the  seventeenth  century 
several  works  appeared  with  this  avowed  purpose. 
Such  was  one  published  anonymously  in  Rouen  in 
1628,  "  Recreations  math6matiques  composers  de 
plusieurs  problemes  d'Arithmetique,  etc."  Schwen- 
ter's  "  Deliciae  Physiko-Mathematicae  oder  mathema- 
tische  und  physikalische  Erquickstunden "  (Altdorf, 
1636)  was  another.  Perhaps  the  best  known  was 
Bachet  de  Meziriac's  "  Problemes  plaisants  et  delect- 
ables,"  which  appeared  in  1612*  the  source  of  several 
of  the  problems  which  still  float  around  our  lower 
schools. 

As  a  quickener  of  the  wit  —  Closely  allied  to  one  or 
two  of  the  reasons  already  mentioned  is  the  idea  that 
arithmetic  is  especially  fitted  to  make  one  sharp, 
keen,  quick-witted.  This  was  one  of  the  leading 
reasons  in  certain  of  the  cloister  schools,  the  subject 
being  there  taught  for  its  bearing  upon  the  training 
of  the  clergy  in  disputation.  Hence  arose  a  mass 
of  catch-problems,  problems  intended  for  argument, 
problems  containing  some  trick  of  language,  etc. 
Such  is  the  famous  one  of  the  widow  to  whom  the 

1  Fifth  edition,  Paris,  1884. 


1 6     THE  TEACHING  OF  ELEMENTARY  MATHEMATICS 

dying  husband  left  two-thirds  of  his  property  if  the 
posthumous  child  should  be  a  girl,  and  one-third  if  it 
should  be  a  boy,  the  remainder  in  either  case  to  the 
child;  the  widow  giving  birth  to  twins,  one  of  each 
sex,  required  to  divide  the  property.  This  particular 
problem  appeared  in  a  collection  of  about  1000  A.D., 
and  is  traced  back  even  to  Hadrian's  time  and  the 
schools  of  law.1  The  title  of  Alcuin's  (735-804)  book, 
"  Propositiones  ad  acuendos  iuvenes,"  and  of  Recorde's 
"The  Whetstone  of  Witte"  (1557)  show  that  for  the 
space  of  nearly  a  thousand  years  these  problems  which 
were  largely  the  product  of  "the  empty  disputations 
and  the  vain  subtleties  of  the  schoolmen"  had  their 
strong  advocates. 

In  the  eighteenth  century,  when  the  reasons  for 
teaching  the  subject  began  to  be  considered  more 
scientifically,  this  idea  was  brought  prominently  to  the 
front  by  a  number  of  leaders  of  educational  thought. 
Thus  Hiibsch,  who  certainly  deserves  to  rank  among 
these  leaders,  remarks  that  "  arithmetic  is  like  a  whet- 
stone, and  by  its  study  one  learns  to  think  distinctly, 
consecutively,  and  carefully."2 

This  is  still  thought  by  certain  conscientious  teachers 
to  be  the  end  in  view  in  teaching  arithmetic.  This 
being  postulated,  they  seek  to  make  arithmetical 
reasoning  unnecessarily  obscure  and  difficult,  allow- 
ing the  use  of  no  equation  forms,  however  simple  and 

1  Cantor,  I,  p.  523,  788.  2  Arithmetica  portensis,  1748. 


HISTORICAL  REASONS  FOR  TEACHING  ARITHMETIC     17 

helpful.  They  simply  conceal  the  equation  in  a  mass 
of  words,  and  cut  off  the  direct  path  for  the  sake  of 
the  exercise  derived  from  stumbling  over  a  circuitous 
route.  This  appears  in  the  subject  of  compound  pro- 
portion and  in  certain  methods  of  treating  percentage. 
The  argument  upon  this  point  of  making  arithmetic 
unnecessarily  hard,  begun  in  Germany  over  a  cen- 
tury ago,1  is,  if  we  may  judge  by  recent  American 
and  German  text-books,  coming  to  a  settlement 
in  two  countries  at  least.  England,  more  conserva- 
tive, and  France,  less  open  minded  in  her  lower 
schools,  still  attempt  to  draw  a  rigid  line  between 
algebra  and  arithmetic,  thus  perpetuating  the  diffi- 
culties of  the  latter. 

Scientific  investigation  of  reasons  —  About  the  close 
of  the  eighteenth  century  the  reasons  for  studying 
mathematics  began  to  be  more  scientifically  considered. 
The  necessity  for  the  subject  in  the  training  of  all 
classes  of  people  began  to  be  generally  recognized. 
Arithmetic  now  began  to  be  looked  upon  as  a  subject 
not  for  the  scientist  and  the  merchant  only,  but  for  the 
soldier,  the  priest,  the  laborer,  the  lawyer,  and  generally"" 
»for  men  in  all  walks  of  life,  and  a  subject  valuable  in 
^various  ways  in  the  mental  equipment  of  the  youth.2 
It  was  to  train  for  business,  but  not  that  alone ;  to  be 

1  Unger,  p.  163. 

2  The  reasons  as  then  considered  are  set  forth  by  Murhard,  System  der 
Elemente  (1798),  quoted  at  length  by  Unger,  p.  142  seq. 

C 


1 8     THE  TEACHING  OF  ELEMENTARY  MATHEMATICS 

interesting,  but  not  that  alone;  to  train  the  child  to 
accuracy,  to  correlate  with  other  subjects,  to  pave  the 
way  for  science,  but  none  of  these  alone.  The  devel- 
opment and  strengthening  of  the  mental  powers  in 
general,  this  was  Pestalozzi's  broad  view  of  the  aim 
in  teaching  arithmetic.  "  So  teach  that  at  every  step 
the  self-activity  of  the  pupil  shall  be  developed,"  was 
Diesterweg's  counsel.1 

Thus  with  the  nineteenth  century  the  self-activity 
and  independence  of  the  pupil  come  to  the  front  in 
education.  The  atmosphere  begins  to  clear.  Out  of 
the  many  reasons  for  the  study  of  arithmetic  two  for- 
mulate themselves  as  prominent,  reasons  as  yet  hidden 
from  the  mechanical  teacher,  who  is  content  with  an 
answer  reached  by  some  mere  rule  of  memory  and  with 
the  recital  of  a  few  score  of  ill-understood  definitions  or 
useless  principles,  but  reasons  which  are  leavening  the 
mass  and  which  will  give  us  vastly  improved  work  in 
the  next  generation. 

1  Diesterweg  and  Heuser's  Methodisches  Handbuch  fur  den  Gesammt- 
unterricht  im  Rechnen,  3  Aufl.,  1839. 


CHAPTER   II 

WHY  ARITHMETIC  is  TAUGHT  AT  PRESENT 

Two  general  reasons  —  In  Chapter  I  a  brief  survey 
of  the  evolution  of  the  reasons  for  teaching  arithmetic 
has  been  given.  It  has  there  appeared  that  it  is 
not  at  all  settled  that  the  subject  should  have  the 
time  now  assigned  it  in  the  curriculum,  or  that  it 
should  be  taught  for  the  purpose  now  in  view,  or  (as 
a  consequence)  that  it  should  be  taught  as  we  now 
teach  it. 

When  we  come  to  examine  the  question  of  the  real 
reason  for  the  study  of  mathematics  to-day,  we  find 
that  we  seek  a  receding  and  an  intangible  something 
which  quite  baffles  our  attempts  at  capture.  Indeed, 
we  may  rather  congratulate  ourselves  that  this  is  the 
case,  and  say  with  one  of  our  contemporary  educators, 
"  For  one,  I  am  glad  we  cannot  express  either  quanti- 
tatively or  qualitatively  the  precise  educational  value 
of  any  study."1 

In  a  general  way,  however,  we  may  summarize  the 
reasons  which  to  the  world  seem  valuable,  by  saying 

1  Hill,  F.  A.,  The  Educational  Value  of  Mathematics,  Educational 
Review,  IX,  p.  349. 

19 


2O     THE  TEACHING  OF  ELEMENTARY  MATHEMATICS 

that  arithmetic,  like  other  subjects,  is  taught  either 
(i)  for  its  utility,  or  (2)  for  its  culture.1  Under  the 
former  is  included  the  general  "  bread-and-butter  value  " 
of  the  subject  and  its  applications ;  under  the  latter, 
its  training  in  logic,  its  bearing  upon  ethical,  religious, 
and  philosophical  thought. 

No  one  will  deny  that  arithmetic  is  taught  for  these 
two  reasons.  It  has  a  bread-and-butter  value  because 
we  need  it  in  daily  life,  in  our  purchases,  in  comput- 
ing our  income,  and  in  our  accounts  generally.  It 
has  a  culture  value  because,  if  rightly  taught,  it  trains 
one  to  think  closely  and  logically  and  accurately. 

The  utility  of  arithmetic  overrated  —  Since  the 
school  requires  the  pupil  to  spend  eight  or  nine  years 
in  studying  arithmetic,  the  general  impression  seems 
to  be  that  this  is  because  arithmetic  is  so  useful  as  to 
demand  so  great  an  expenditure  of  time.  This  view 
cannot,  however,  be  justified.  "The  direct  utilitarian 
value  of  arithmetic  —  its  value  to  the  breadwinner 
—  has  been  much  overestimated;  or,  perhaps,  it 
is  nearer  the  truth  to  say  that,  while  accuracy  and 

1  Fitch,  Lectures  on  Teaching,  6th  ed.,  1884,  chaps.  x,xi;  Payne's  trans, 
of  Compayre's  Lectures  on  Pedagogy,  p.  379 ;  Reidt,  F.,  Anleitung  zum 
mathem.  Unterricht,  Berlin,  1886,  p.  101  ;  Fitzga,  E.,  Die  natiirliche 
Methode  des  Rechen-Unterrichtes,  I.  Theil,  Wien,  1898,  p.  44,  hereafter 
referred  to  as  Fitzga ;  Stammer,  Ueber  den  ethischen  Wert  des  mathemat. 
Unterrichts,  in  Hoffmann's  Zeitschrift,  XXVIII,  p.  487,  and  other  articles 
in  this  journal.  The  best  of  the  recent  discussions  is  given  in  Knilling, 
R.,  Die  naturgemasse  Methode  des  Rechen-Unterrichts  in  der  deutschen 
Volksschule,  II.  Teil,  Munchen,  1899. 


WHY  ARITHMETIC  IS  TAUGHT  AT  PRESENT          21 

speed  in  simple  fundamental  processes  have  been 
underestimated,  the  value  of  presenting  numerous  and 
varied  themes  in  pure  arithmetic,  and  of  pressing  each 
to  great  and  difficult  lengths,  has  been  seriously  over- 
rated."1 

For  the  ordinary  purposes  of  non-technical  daily 
life  we  need  little  of  pure  arithmetic  beyond  (i)  count- 
ing, the  knowledge  of  numbers  and  their  representa- 
tion to  billions  (the  English  thousand  millions),  (2) 
addition  and  multiplication  of  integers,  of  decimal  frac- 
tions with  not  more  than  three  decimal  places,  and 
of  simple  common  fractions,  (3)  subtraction  of  inte- 
gers and  decimal  fractions,  and  (4)  a  little  of  division. 
Of  applied  arithmetic  we  need  to  know  (i)  a  few 
tables  of  denominate  numbers,  (2)  the  simpler  prob- 
lems in  reduction  of  such  numbers,  as  from  pounds 
to  ounces,  (3)  a  slight  amount  concerning  addition 
and  multiplication  of  such  numbers,  (4)  some  simple 
numerical  geometry,  including  the  mensuration  of  rec- 
tangles and  parallelepipeds,  and  (5)  enough  of  per- 
centage to  compute  a  commercial  discount  and  the 
simple  interest  on  a  note. 

The  table  of  troy  weight,  for  example,  forms  part 
of  the  technical  education  of  the  goldsmith,  the  tables 
of  apothecaries'  measures  form  part  of  the  technical 
education  of  a  drug  clerk  or  a  physician,  equation  of 
payments  may  have  place  in  the  training  of  a  few 

1  Hill,  F.  A.,  in  Educational  Review,  IX,  p.  350. 


22     THE  TEACHING  OF   ELEMENTARY  MATHEMATICS 

bookkeepers,  but  for  the  great  mass  of  people  these 
time-consuming  subjects  have  no  bread-and-butter 
value.  How  many  business  men  have  any  more 
occasion  to  use  the  knowledge  of  series  which  they 
may  have  gained  in  school,  than  to  use  the  differen- 
tial calculus?  The  same  question  may  be  asked  con- 
cerning cube  root,  and  even  concerning  square  root; 
most  people  who  have  occasion  to  extract  these  roots 
(engineers  and  scientists)  employ  tables,  the  cumber- 
some method  of  'the  text-book  having  long  since  passed 
from  their  minds.  A  like  question  might  be  raised 
respecting  alligation,  only  this  has  happily  nearly  dis- 
appeared from  American  arithmetics,  although  it  still 
remains  a  favorite  topic  in  Germany.  Equation  of 
payments,  compound  interest  (as  taught  in  school), 
compound  (and  even  simple)  proportion,  greatest  com- 
mon divisor,  complex  fractions,  and  various  other 
chapters  are  open  to  the  same  inquiry.  These  sub- 
jects, which  are  the  ones  which  consume  most  of  the 
time  in  the  arithmetic  classes  of  the  grades  after  the 
fourth,  are  so  rarely  used  in  business  that  the  ordi- 
nary tradesman  or  professional  man  almost  forgets 
their  meaning  within  a  few  months  after  leaving 
school. 

Of  compound  numbers,  which  occupy  a  year  of 
the  pupil's  time  in  school  (a  year  saved  in  most 
civilized  countries  except  the  Anglo-Saxon,  by  the 
use  of  the  metric  system),  the  amount  actually  needed 


WHY  ARITHMETIC  IS  TAUGHT  AT  PRESENT          23 

in  daily  life  is  very  slight.  The  common  measures 
of  length,  of  area,  of  volume  (capacity),  and  of  avoir- 
dupois weight  are  necessary.  One  also  needs  to  be 
able  to  reduce  and  to  add  compound  numbers,  but 
rarely  those  involving  more  than  two  or  three  de- 
nominations. For  practical  purposes  a  problem  like 
the  following  is  useless :  Divide  2  Ibs.  7  oz.  19  pwt. 
by  5  oz.  6  pwt.  12  gr. 

Most  of  the  problems  of  common  fractions  are  very 
uncommon.  In  business  and  in  science,  common  frac- 
tions with  denominators  above  100  are  rare,  the  deci- 
mal fraction  (which  has  now  become  the  "common" 
one)  being  generally  used. 

What,  then,  should  be  expected  of  a  child  in  the 
way  of  the  utilities  of  arithmetic?  (i)  A  good  work- 
ing knowledge  of  the  fundamental  processes  set  forth 
on  p.  21 ;  (2)  accuracy  and  reasonable  rapidity,  sub- 
jects which  will  be  discussed  later  in  this  work ;  and 
(3)  a  knowledge  of  the  ordinary  problems  of  daily 
life.  Were  arithmetic  taught  for  the  utilities  alone, 
all  this  could  be  accomplished  in  about  a  third  of  the 
time  now  given  to  the  subject. 

The  culture  value  —  Although  it  is  true  that  a  large 
part  of  our  so-called  applied  or  practical  arithmetic  is 
not  generally  applicable  to  ordinary  business,  and 
hence  is  quite  impractical,  it  by  no  means  follows 
that  it  may  not  serve  a  valuable  purpose.  "  Hamlet " 
may  bring  us  neither  food  nor  clothing,  and  yet  a 


24     THE  TEACHING  OF  ELEMENTARY  MATHEMATICS 

knowledge  of  Shakespeare's  masterpiece  is  valuable 
to  every  one.  It  is  a  matter  of  no  moment  in  the 
business  affairs  of  most  men  that  they  know  where 
the  Caucasus  Mountains  are,  or  which  way  the  Rhine 
flows,  or  who  Cromwell  was,  and  yet  we  cannot 
afford  to  be  ignorant  of  these  facts. 

How,  then,  can  the  teaching  of  arithmetic  beyond 
the  mere  elements  be  justified?  Fitch,  in  his  "  Lectures 
on  Teaching,"  already  cited,  puts  the  case  tersely.  He 
says,  "  Arithmetic,  if  it  deserves  the  high  place  that 
it  conventionally  holds  in  our  educational  system, 
deserves  it  mainly  on  the  ground  that  it  is  to  be 
treated  as  a  logical  exercise"  Bain  remarks  in  the 
same  tenor:  "All  this  presupposes  mathematics  in 
its  aspect  of  training;  or,  as  providing  forms,  meth- 
ods, and  ideas,  that  enter  into  the  whole  mechanism 
of  reasoning,  wherever  that  takes  a  scientific  shape. 
As  culture  imposed  upon  every  one,  this  is  its  highest 
justification.  But,  if  so,  these  fruitful  ideas  should  be 
made  prominent  in  teaching ;  that  is,  the  teacher  should 
be  conscious  of  their  all-penetrating  influence.  More- 
over, he  should  keep  in  view  that  nine-tenths  of  pupils 
derive  their  chief  benefit  from  these  ideas  and  forms 
of  thinking  which  they  can  transfer  to  other  regions 
of  knowledge ;  for  the  large  majority  the  solution  of 
problems  is  not  the  highest  end."  1 

1  Bain,  A.,  Education,  p.  152.  See  also  Fitzga,  p.  27 ;  Rein,  Pickel 
and  Scheller,  Theorie  und  Praxis  des  Volksschulunterrichts,  I,  p.  350. 


WHY  ARITHMETIC  IS  TAUGHT  AT  PRESENT          25 

In  other  words,  it  seems  advisable  to  give  the  child 
some  training  in  logic.  But  logic  as  a  science  is  too 
abstract  for  him.  Hence  the  school  substitutes  that 
subject,  which,  at  the  time,  offers  the  best  oppor- 
tunity for  this  training.  This  is  the  more  valuable, 
in  that  there  is  incidentally  accomplished  another 
result,  the  keeping  of  the  numerical  machinery  in  use 
while  the  child  is  in  school,  so  that  his  powers  of  cal- 
culating will  be  unimpaired  from  inactivity  when  he 
leaves.  Arithmetic  is  well  chosen  for  this  training 
in  logic,  because  it  furnishes  almost  the  only  example 
of  an  exact  science  below  the  high  school,  as  the 
American  courses  are  usually  arranged.  And  although 
induction  is  more  valuable  to  the  child  than  deduction, 
and  while  it  must  be  the  keynote  of  primary  arithmetic, 
deduction  plays  an  important  part  in  the  latter  portion 
of  the  subject.  The  fact  that  the  child  finds  a  posi- 
tive truth,  an  immutable  law,  at  the  time  in  his  develop- 
ment when  he  is  naturally  filled  with  doubt,  with  the 
desire  to  investigate,  and  with  the  feeling  that  he 
must  put  away  childish  things,  has  a  value  difficult 
properly  to  appreciate.  He  is  not  sure  that  every 
flower  has  petals,  that  every  animal  needs  oxygen, 
that  "  most  unkindest "  is  bad  grammar,  or  that 
Columbus  was  the  real  discoverer  of  America ;  but 
he  is  sure,  and  no  argument  can  shake  his  faith,  that 
whatever  may  happen  to  the  universe  in  which  he 
lives,  (a  -f  6)*  will  always  equal  a*  +  2  ab  -f  #, 

< 


26     THE  TEACHING  OF  ELEMENTARY  MATHEMATICS 

So  arithmetic  may,  even  by  obsolete  problems,  train 
the  mind  of  the  child  logically  to  attack  the  every-day 
problems  of  life.  If  he  has  been  taught  to  think  in 
solving  his  school  problems,  he  will  think  in  solving 
the  broader  ones  which  he  must  thereafter  meet. 
The  same  forms  of  logic,  the  same  attention  to  detail, 
the  same  patience,  and  the  same  care  in  checking 
results  exercised  in  solving  a  problem  in  greatest 
common  divisor,  may  show  itself  years  later  in  com- 
merce, in  banking,  or  in  one  of  the  learned  profes- 
sions. Hence,  arithmetic,  when  taught  with  this  in  mind, 
gives  to  the  pupil  not  knowledge  of  facts  alone,  but 
that  which  transcends  such  knowledge,  namely,  power. 

It  must  not,  however,  be  thought  from  its  name  that 
this  culture  phase  of  the  subject  is  of  value  only  as 
a  luxury,  like  the  ability  to  dabble  in  music  or  paint- 
ing. Just  because  it  is  the  child  of  the  man  in  poor 
or  moderate  circumstances  who  must  make  his  own 
way  in  the  world,  it  is  for  the  common  people  that 
this  culture  phase  is  most  valuable. 

Teachers  generally  fail  here  —  The  lower  elementary 
teacher  of  arithmetic  is  usually  more  successful  than 
the  one  in  the  higher  grades.  There  are  several 
reasons  for  this  —  the  primary  part  of  the  subject 
has  been  much  better  investigated,  better  books  have 
been  written  about  it,  good  higher  arithmetics  are 
rare,  and  the  child  in  the  lower  grades  has  not  to 
face  the  nervous  shock  which  comes  a  little  later; 


WHY  ARITHMETIC  IS  TAUGHT  AT  PRESENT          27 

but  one  of  the  chief  reasons  is  that  the  primary 
teacher  knows  why  she  is  teaching  arithmetic,  while 
often  the  one  in  the  higher  grades  does  not.  In  the 
first  grade  the  subject  is  being  taught  largely  for  its 
utilities,  and  induction  plays  the  important  part;  this 
the  teacher  knows  and  hence  she  succeeds.  In  the 
seventh  grade  the  teacher  is  apt  to  think  that  induc- 
tion still  plays  the  leading  r61e,  an  error  which  gives 
rise  to  much  poor  teaching. 

Recognition  of  the  culture  value  —  This  culture  value 
is  brought  out  first  by  letting  the  amount  taken  on 
authority  of  the  book  or  the  teacher  be  a  minimum. 
"In  education  the  process  of  self-development  should 
be  encouraged  to  the  uttermost.  Children  should  be 
led  to  make  their  own  investigations  and  to  draw  their 
own  inferences.  They  should  be  told  as  little  as 
possible,  and  induced  to  discover  as  much  as  possi- 
ble. .  .  .  Any  piece  of  knowledge  which  the  pupil 
has  himself  acquired,  any  problem  which  he  has 
himself  solved,  becomes  by  virtue  of  the  conquest 
much  more  thoroughly  his  than  it  could  else  be."1 

This  is  not  to  be  construed  to  mean  that  nothing 
is  to  be  taken  for  granted.  We  must  assume,  for 
example,  that  equals  result  from  adding  equals  to 
equals.  But  when  Euclid  was  criticised  for  proving 
that  one  side  of  a  triangle  is  less  than  the  sum  of 
the  other  two,  as  having  proved  what  even  the  beasts 

1  Spencer,  Education. 


28     THE  TEACHING  OF  ELEMENTARY  MATHEMATICS 

know,  his  disciples  were  entirely  right  in  saying  that 
they  were  not  merely  teaching  facts,  but  were  en- 
gaged in  the  far  more  important  work  of  giving  the 
power  to  prove  the  facts.  As  Bain  puts  it,  referring 
to  the  higher  grades,  "The  pupil  should  be  made 
to  feel  that  he  has  accepted  nothing  without  a  clear 
and  demonstrative  reason,  to  the  entire  exclusion  of 
authority,  tradition,  prejudice,  or  self-interest." 1 
-  What,  then,  shall  be  said  of  text-books  which  give 
long  lists  of  "  Principles "  as  a  kind  of  inspired  reve- 
lation to  pupils  ?  So  far  as  these  are  statements  of 
business  customs  they  have  place;  but  they  are  gener- 
ally theorems,  capable  of  easy  proof,  and  of  no  great 
value  without  this  proof. 

Furthermore,  if  we  would  make  a  clear  thinker 
of  the  pupil,  he  should  not  be  compelled  to  learn, 
verbatim,  all  or  even  a  majority  of  the  definitions  of 
the  text-book.  This  does  not  exclude  those  which 
are  true  and  understandable  and  valuable  in  subse- 
quent work;  but  it  refers  to  those  which  are  false, 
unintelligible,  and  not  usable,  and  to  partial  definitions 
in  all  cases  where  the  memorizing  of  the  same  hinders 
the  comprehension  of  the  complete  definition  subse- 
quently. For  example,  what  teacher  of  arithmetic  can 
define  number  in  such  way  as  to  have  the  definition 
both  true  and  intelligible  to  young  pupils,  those  below 
the  high  school?  And  if  he  could  do  so,  of  what 

1  Education,  p.  149. 


WHY  ARITHMETIC  IS  TAUGHT  AT  PRESENT          29 

value  would  it  be  ?  Or  who  would  care  to  undertake 
the  definition  of  quantity?1  The  fact  is  that  the 
simpler  the  term  the  more  difficult  the  definition. 
Since  a  definition  must  explain  terms  by  the  use  of 
terms  more  simple,  it  follows  that  one  must  sometime 
come  to  terms  incapable  of  definition.2  In  daily  life 
we  do  not  learn  definitions  verbatim ;  if  asked  to 
define  horse,  the  definition  would  probably  include  the 
mule  and  zebra  and  numerous  others  of  the  equine 
family.  The  usual  definition  of  multiplication  has 
hindered  the  work  of  many  a  child  in  fractions,  and 
yet,  even  in  the  first  grade  he  multiplies  by  the  frac- 
tion J.  While  it  is  true  that  partial  truths  precede 
complete  ones,  it  is  poor  teaching  to  impress  this  partial 
truth  on  the  mind  so  indelibly,  by  a  memorized  state- 
ment, as  to  make  the  complete  truth  difficult  of  as- 
similation. For  example,  a  teacher  drills  a  class  to 
memorize  the  fiction  that  if  the  second  term  of  a 
proportion  is  less  than  the  first,  the  fourth  must  be 
less  than  third, — a  statement  entirely  unnecessary  in 
the  logical  treatment  of  proportion,  and  then,  when 
the  pupils  come  to  meet  i  :  —  2  =  —  2  : 4,  they  are  lost. 
To  test  the  matter  a  little  further,  let  any  reader 

1  Those  who  may  be  ambitious  to  make  the  attempt  might  first  read 
Laisant,  La  Mathematique,  Paris,  1898,  p.  14,  hereafter  referred  to  is  Laisant, 
or  the  simple  definition  of  number  in  the  Encyklopadie  der  mathematischen 
Wissenschaften,  I.  Heft,  Leipzig,  1898,  now  in  process  of  publication. 

1  Duhamel,  J.-M.-C,  Des  Methodes  dans  les  Sciences  de  Rai- 
sonnement.  !**«  partie,  3»*me  ed.,  Paris,  1885,  p.  1 6. 


30     THE  TEACHING  OF  ELEMENTARY  MATHEMATICS 

repeat  the  definition  of  number,  as  it  was  once  burnt 
into  his  memory,  and  see  if  TT(=  3.14159  •••)  is  a 
number  according  to  this  definition,  —  or  V2,  or  V—  i. 
Or  try  the  definition  of  arithmetic  and  see  if,  by  this 
statement,  the  table  of  avoirdupois  weight  is  any  part 
of  the  subject.  Does  the  definition  of  multiplication, 
as  usually  memorized,  cover  even  the  simple  case  of 
f  x  f ,  to  say  nothing  of  A/2  x  "\/3  or  —  V—  i  x  V—  3  ? 
By  the  common  definition  of  factor  is  \  a  factor  of  J  ? 
By  the  definition  of  square  root,  as  usually  learned, 
have  we  any  right  to  speak  of  the  square  root  of  3, 
since  3  has  not  two  equal  factors?  Are  our  arith- 
metics clear  enough  in  statement  so  that  the  memoriz- 
ing of  their  definitions  will  tell  a  pupil  whether  the 
simple  series  2,  2,  2,  2,  •••  is  an  arithmetical  or  a 
geometric  progression,  or  neither? 

The  old  argument  that  learning  definitions  strengthens 
the  memory  and  gives  a  good  vocabulary,  has  too  few 
advocates  now  to  make  it  worth  consideration.  "The 
r61e  of  the  memory,  certainly  necessary  in  matters 
mathematical  as  elsewhere,  should  be  reduced  in  a 
general  way  to  very  limited  proportions  in  rational 
teaching.  It  is  not  the  images,  the  figures,  or  the 
formulae  which  must  be  impressed  upon  the  mind,  so 
much  as  it  is  the  power  of  reasoning."1 

1  "  Ce  ne  sont  pas  les  images,  figures  ou  formules,  dont  U  faut  surtout 
laisser  1'empreinte  dans  le  cerveau;  c'est  la  faculte  du  raisonnement." 
Laisant,  p.  191. 


WHY  ARITHMETIC  IS  TAUGHT  AT  PRESENT          31 

This  opposition,  on  the  part  of  leaders  in  education, 
to  the  burdening  of  children's  memories,  is  not  new. 
Locke  voiced  the  same  sentiment :  "  And  here  give  me 
leave  to  take  notice  of  one  thing  I  think  a  fault  in  the 
ordinary  method  of  education ;  and  that  is,  the  charging 
of  children's  memories,  upon  all  occasions,  with  rules 
and  precepts,  which  they  often  do  not  understand,  and 
constantly  as  soon  forget  as  given."  l  "  Teachers  at 
one  time  believed  that  the  first  object  of  primary 
instruction  is  to  cultivate  the  verbal  memory  of  their 
pupils,  when,  in  fact,  the  verbal  memory  is  one  of  the 
few  faculties  of  our  nature  which  need  no  cultivation."  2 
Of  the  two,  to  learn  all  of  the  definitions  of  a  text-book 
or  none,  the  latter  plan  is  unquestionably  the  better. 

But  while  memorized  definitions  may  not  unfrequently 
be  justified,  this  is  rarely  true  of  the  memorized  rule. 
The  glib  recitation  of  rules  for  long  division,  greatest 
common  divisor,  etc.,  which  one  hears  in  some  schools 
—  what  is  all  this  but  a  pretence  of  knowledge?  "If 
learning  is  a  process  of  gaining  knowledge,  that  is, 
a  true  apprehension  of  realities,  it  excludes  verbal  mem- 
orizing, cramming,  and  everything  that  resolves  itself 
on  close  scrutiny  into  a  pretence  of  knowledge  getting."  3 

But  not  only  is  this  old-fashioned  rule-learning  (un- 
happily not  yet  extinct)  a  sham ;  it  is  wholly  unscientific. 
Tillich,  one  of  the  best  teachers  of  arithmetic  of  the 

1  On  Education,  Daniel's  edn.,  p.  126.  *  Tate. 

*  Dr.  James  Sully,  in  the  Educational  Times,  December,  1890. 


32     THE  TEACHING  OF  ELEMENTARY  MATHEMATICS 

first  half  of  the  nineteenth  century,  saw  the  danger  of 
dogmatic  rules.  "It  is,"  he  said,  "just  as  unpsycho- 
logical  to  begin  the  teaching  of  arithmetic  by  a  mass 
of  inherited  rules  as  it  is  senseless  to  try  to  teach  lan- 
guage to  children  by  means  of  mere  rules  of  speech. 
.  .  .  Since  these  rules  were  not  independently  worked 
out  by  the  child,  but  are  simply  the  memorized  results 
of  others'  work,  it  cannot  but  be  true  that  the  arith- 
metic of  most  of  the  pupils  is  a  mere  mechanism,  and 
a  distasteful  one  at  that."  l  So,  too,  Jean  Mac6,  in  his 
well-known  "Arithmetic  of  a  Grand-Papa,"  remarks 
that  to  have  a  child  begin  with  the  abstract  rule,  follow- 
ing this  by  the  solution  of  a  lot  of  problems,  is  to  com- 
pletely reverse  the  order  of  human  development.2 

There  are,  however,  a  few  rules  of  operation  which 
must  be  learned  for  the  sake  of  facility  and  speed  in 
numerical  calculation.  Such  is  the  rule  for  substituting 
another  and  a  simpler  operation  for  that  of  dividing  one 
fraction  by  another.  But  this  does  not  mean  that  such 
a  rule  is  to  be  given  as  a  kind  of  inspired  dogma.  It 
is  quite  as  easy,  and  far  more  valuable,  to  lead  the 
child  to  discover  it  for  himself.  Even  as  far  back  as 
Roger  Ascham  this  was  realized,  though  seldom  prac- 
tised. "We  do  not  contemne  rewles,"  said  he,  "but 

1  Lehrbuch  der  Arithmetik,  p.  xi.     In  a  similar  line,  Reidt,  Fr.,  An- 
leitung  zum  mathematischen  Unterricht  an  hoheren  Schulen,  Berlin,  1886, 
p.  103. 

2  L'Arithmetique  du  Grand-Papa,  4^me  £d.,  p.  12. 


WHY  ARITHMETIC  IS  TAUGHT  AT  PRESENT          33 

we  gladly  teach  rewles ;  and  teach  them  more  plainlie, 
sensiblie,  and  orderlie  than  they  be  commonlie  taught 
in  common  scholes."  1  And  the  best  of  summaries  of 
method  that  has  recently  appeared  asserts :  "  Whoever 
would  bring  his  pupils  to  intelligent  computation  (zu 
einen  verstandnisvollen  Rechnen)  should  develop  no 
rule,  but  should  wait  until  the  children  themselves  dis- 
cover it  (bis  die  Kinder  selbst  darauf  kommen)."  2 

Aside  from  the  fact  that  we  make  almost  no  use 
of  the  rules  of  operation  in  our  daily  computations, 
needing  but  a  few  rules  of  business  and  theorems  of 
mensuration,  there  is  the  further  consideration  that 
the  child  does  not  like  to  solve  by  rule.  To  use  his 
common  sense  is  to  become  a  discoverer,  and  the 
zeal  for  discovery  is  one  of  the  inborn  traits  of  the 
human  mind.  If  all  mathematical  problems  were 
solved,  or  if  we  had  rules  for  solving  them,  all  inter- 
est in  the  subject  would  vanish. 

Of  course  the  same  objection  which  exists  as  to 
rules  exists  in  even  greater  measure  as  to  undemon- 
strated  formulae,  which  are  merely  rules  put  in  un- 
familiar language.  To  fill  the  child's  mind  with  a 
list  of  formulae  for  percentage,  for  example,  is  to 
take  a  human  soul  and  try  to  make  a  machine  of 
it.  "  If  one  learns  only  by  memory,  and  does  not 
think,  all  remains  dark."8 

What,   then,  shall  be   said   of  the  educative  value 

1  The  Scholemaster.  2  Fitzga,  p.  48.  »  Confucius. 

D 


34     THE  TEACHING  OF  ELEMENTARY   MATHEMATICS 

of  the  old-fashioned  arithmetic  which  put  its  prob- 
lems in  "cases,"  each  preceded  by  the  rule?  Surely 
a  more  mechanical  device  could  hardly  be  invented. 
And  yet  these  books  exist  to-day  in  thousands  of 
schools  in  England  and  America.  And  if  it  be  said 
that  these  books  in  the  schools  of  fifty  years  back 
produced  good  arithmeticians,  let  it  not  be  forgotten 
that  far  more  time  was  then  given  to  the  subject. 
Good  arithmeticians  were  produced  in  spite  of,  not 
because  of,  such  books. 

What  chapters  bring  out  the  culture  value — It  is  not 
so  much  the  particular  chapter  as  the  way  it  is  taught 
that  brings  out  the  educational  value  of  arithmetic. 
A  person  may  have  exercise  in  logic  by  studying  alli- 
gation—  merely  indeterminate  equations  in  an  awk- 
ward mediaeval  form.  But  the  best  results  will  naturally 
come  from  those  parts  that  appeal  to  the  child's  life 
and  interests. 

For  example,  longitude  and  time,  a  subject  with 
but  slight  utilitarian  value  to  most  people,  may  be 
so  taught  as  to  have  high  culture  value.  The  inter- 
est attaching  to  the  "date  line"  and  to  the  recent 
world-movement  of  "standard  time,"  renders  the  sub- 
ject a  delightful  one  to  children  of  a  certain  age. 
But  its  value  is  lost  when  a  book  gives  the  form 
"75°  -*-  15  =  5  hrs.,"  since  it  destroys  the  child's  pre- 
conceived and  correct  ideas  of  the  nature  of  division ; 
accuracy  of  statement  and  of  thought  have  been 


WHY  ARITHMETIC  IS  TAUGHT  AT  PRESENT          35 

sacrificed  for  a  mere  answer,  an  arithmetical  birth- 
right sold  for  a  mess  of  pottage. 

Similarly,  "  true  discount "  may  be  made  interest- 
ing, and  the  reasoning  may  give  rise  to  logical  power. 
But  this,  like  other  subjects  that  at  once  occur  to 
the  teacher,  is  open  to  the  fatal  objection  that  it 
gives  a  wrong  idea  of  business.  However  much  the 
pupil  may  be  warned,  the  name  "true  discount"  will 
cling  to  him,  and  he  must  learn,  after  his  ^hool 
days  have  gone  by,  that  the  true  is  really  the  false 
discount  in  the  life  he  is  to  live. 

What  may  well  be  omitted  —  In  considering  what 
may  profitably  be  omitted  from  the  arithmetic  of  to- 
day, there  is,  of  course,  the  bugbear  of  the  examina- 
tion to  be  taken  into  account  as  a  practical  question. 
But  looking  at  the  subject  from  the  standpoint  of 
the  educator  rather  than  the  coach,  we  have  to  con- 
sider what  there  is  that  appeals  neither  to  the 
utilitarian  nor  to  the  culture  value,  or  that  is  found 
wanting  for  other  reasons. 

i.  The  following  may  be  said  to  have  little  or  no 
utilitarian  value  for  the  general  citizen,  and  because 
they  give  a  false  notion  of  business  they  may  also 
be  rejected  as  undesirable  exercises  in  logic. 

(a)  Equation  of  payments. 

(b)  Alligation  (now  rapidly  disappearing  from  Eng- 
lish and  American   text-books,  although  still  found  in 
the  German). 


36     THE  TEACHING  OF  ELEMENTARY  MATHEMATICS 

(c)  Insurance,  in  the  form  usually  presented  in  text- 
books. 

(d)  "  Profit  and  Loss,"  the  text-book  expression  not 
having     the    American     business    meaning,    and    the 
problems  being  merely  ordinary  ones  of   simple   per- 
centage, not  worthy  of  a  special  chapter. 

(e)  Exchange   as   usually   taught.      If    the    modern 
business    problems   are   given,   with    the   modern   ma- 
chinery  for   exchange,   the    subject  is  valuable.      Of 
course   arbitrated   exchange  has   no   value  per  se   for 
the  ordinary  citizen;  it  is  part  of  the  technical  train- 
ing of  a  few  brokers. 

(/)  Commission  and  brokerage  so  far  as  the  sub- 
ject relates  to  problems  like  the  following :  "  A  sends 
B  $1000  with  which  to  buy  wheat  on  a  2^%  com- 
mission :  how  much  can  B  invest  ? " 

(£•)  Stocks,  where  the  problems  require,  as  in 
many  text-books,  fractional  numbers  of  shares,  like 
the  buying  of  8f-  shares,  or  where  they  call  for  un- 
used quotations  like  lOQ^f. 

(fi)  Partial  payments  beyond  the  common  methods 
in  the  state  in  which  the  pupil  lives. 

(i)  Annual  interest,  beyond  the  mere  elements. 

(/)  Compound  interest,  beyond  the  ability  to  find 
such  interest.  The  banker,  of  course,  employs  tables 
whenever  he  has  occasion  to  use  the  subject. 

(k)  Compound  proportion,  a  subject  in  which 
hardly  a  text-book  problem  can  be  found  that  has 


1  UNlVERif 


WHY  ARITHMETIC  IS  TAUGHT  AT  PREEl--      37 

any  practical  value,  in  spite  of  the  pretensions  of  the 
subject.  As  for  mathematical  explanation,  it  would  be 
difficult  to  find  a  text-book  which  makes  any  attempt 
in  that  direction. 

(/)  Problems  in  denominate  numbers  involving 
more  than  three  denominations  at  a  time,  and  those 
involving  tables  not  needed  in  daily  life  —  troy, 
apothecaries',  etc.  Similarly  the  semi-obsolete  meas- 
ures, the  stone  (in  America),  the  barleycorn,  the 
tun,  the  pipe,  etc.,  and  the  technical  measures,  the 
square  (in  shingling),  the  perch,  the  quintal,  etc., 
have  no  place  in  the  common  schools.  There  is, 
indeed,  a  somewhat  serio-comic  aspect  of  the  matter 
as  set  forth  in  the  Football  Field:  "A  gallon  isn't  a 
gallon.  It's  a  wine  gallon,  or  one  of  three  different 
sorts  of  ale  gallon,  or  a  corn  gallon,  or  a  gallon  of 
oil  ;  and  a  gallon  of  oil  means  seven  and  a  half 
pounds  for  train  oil,  and  eight  pounds  for  some 
other  oils.  If  you  buy  a  pipe  of  wine,  how  much 
do  you  get?  Ninety-three  gallons  if  the  wine 
be  Marsala,  ninety-two  if  Madeira,  a  hundred  and 
seventeen  if  Bucellas,  a  hundred  and  three  if  Port,  a 
hundred  if  Teneriffe.  What  is  a  stone?  Fourteen 
pounds  of  a  living  man,  eight  of  a  slaughtered  bul- 
lock, sixteen  of  cheese,  five  of  glass,  thirty-two  of 
hemp,  sixteen  and  three-quarters  of  flax  at  Belfast, 
four  and  twenty  of  flax  at  Downpatrick.  It  is  four- 
teen pounds  of  wool  as  sold  by  the  growers,  fifteen 


38     THE  TEACHING  OF  ELEMENTARY  MATHEMATICS 

pounds  of  wool  as  sold  by  the  wool-staplers  to  each 
other.  .  .  .  Our  very  sailors  do  not  mean  the  same 
thing  when  they  talk  of  fathoms.  On  board  a  man- 
of-war  it  means  six  feet,  on  board  a  merchantman  five 
and  a  half  feet,  on  board  a  fishing  vessel  five  feet."  1 

Of  course  we  may  say  that  in  America  "  we  have 
changed  all  that,"  and  that  we  have  no  such  non- 
sense. And  yet  many  a  school  to-day  teaches  the 
children  the  length  of  the  cubit,  which  nobody  knows 
or  can  know,  because  it  varied,  and  our  various  states 
have  different  laws  and  customs  as  to  what  consti- 
tutes a  bushel  of  grain,  a  perch  of  stone,  etc.,  and 
we  are  quite  as  unsettled  with  respect  to  many  meas- 
ures as  is  Great  Britain. 

"  Of  late  years,  there  has  been  some  reform  in  this 
particular  (the  applications  of  arithmetic),  and  a  few 
of  the  monstrosities  of  the  old  curriculum,  notably 
our  ancient  enemy,  duodecimals,  have  been  thrown 
overboard.  But  there  still  remain  many  things,  as 
taught  in  our  schools,  which  occupy  time  that  could 
better  be  devoted  to  the  study  of  other  subjects,  or 
at  least  to  a  greater  degree  of  practice  in  simple 
operations.  .  .  .  Compound  interest,  compound  pro- 
portion, compound  partnership,  cube  root  and  its 
applications,  equation  of  payments,  exchange,  'similar 
surfaces/  and  the  mensuration  of  the  trapezoid  and 
trapezium,  of  the  prism,  pyramid,  cone,  and  sphere, 

1  Educational  Times,  October,  1892. 


WHY  ARITHMETIC  IS  TAUGHT  AT   PRESENT          39 

are  proposed  to  be  dropped  from  the  course  in   the 
(Boston)  grammar  school."1 

2.  The  following  may  be  said  to  have  some,  and 
might  have  much,  culture  value,  but  should  be 
omitted  on  other  grounds.2 

(a)  Series,    because    the    subject    can    better    be 
treated  where  it  belongs,   in   algebra. 

(b)  The    long    form    of    greatest    common    divisor 
before  about  the  eighth   grade,  because   it  is  taught 
only  for  its  logic,  and  this  logic  is  too  much  for  the 
average  child  below  that  grade. 

(c)  Compound    proportion,    already    mentioned,    be- 
cause almost  no  arithmetic  pretends  to  treat  it  other- 
wise than  by  rule,  and  an  explanation  is  too  difficult 
for  pupils  —  as  apparently  for  authors.      Indeed,  it  is 
doubtful  if  the   child  derives  much   good   even   from 
simple  proportion  as  usually  presented. 

Relative  value  of  culture  and  utility  —  Since  it 
appears  that  arithmetic  is  taught  for  these  two 
general  reasons,  a  question  arises  as  to  their  relative 
importance.  But  this  it  is  impossible  to  answer.  We 
lack  a  unit  of  measure.  Laisant  remarks3  that  it  is 

1  Walker,  F.  A.,  Arithmetic  in  Primary  and  Grammar  Schools,  Boston, 
1887,  p.  12. 

2  "  The  charge  I  make  against  the  existing  course  of  study  is,  that  it  is 
largely  made  up  of  exercises  which  are  not  exercises  in  arithmetic  at  all,  or 
principally,  but  are  exercises  in  logic  ;  and,  secondly,  that,  as  exercises  in 
logic,  they  are  either  useless  or  mischievous?'     Walker,  Ib.,  17. 

8  La  Mathematique,  p.  10. 


40     THE  TEACHING  OF  ELEMENTARY  MATHEMATICS 

like  asking  which  is  the  more  important,  eating  or 
sleeping;  the  loss  of  either  is  fatal.  The  teacher 
who  recognizes  in  the  subject  only  its  applications  to 
trade,  would  better  give  up  teaching;  the  one  who 
sees  in  it  only  an  exercise  in  logic  will  also  fail ; 
but  the  greatest  failure  comes  from  seeing  in  the 
subject  neither  utility  nor  logic,  as  is  the  case  with 
the  teacher  who  blindly  follows  the  old-style,  tradi- 
tional text-book. 

But  what  shall  be  said  for  the  teacher  who  fears 
to  omit  certain  problems  which  are  not  utilitarian  and 
whose  culture  value  is  counterbalanced  by  the  fact 
that  they  give  a  false  notion  of  business,  or  to  omit 
those  traditional  puzzles  which  depend  for  their  diffi- 
culty upon  their  ambiguity  of  statement?  Many  a 
teacher,  especially  in  our  country  schools,  will  confess 
to  such  a  fear  of  omitting  problems,  lest  he  be  ac- 
cused of  inability  to  solve  them.  It  would  be  well 
for  all  teachers  to  assist  in  creating  a  sentiment  in 
favor  of  omitting  the  unquestionably  superfluous  or 
dangerous,  and  thus  to  avoid  this  weak  criticism. 
It  should  also  be  understood  by  timid  teachers  that 
it  is  no  disgrace  to  be  unable  to  solve  every  puzzle 
that  may  be  sent  in,  or  even  every  legitimate  problem. 
And  for  those  who  may  feel  inclined  to  boast  that  they 
have  never  seen  a  problem  in  arithmetic  which  they 
could  not  solve,  it  may  be  interesting  and  instructive  to 
attempt  to  prove  the  following  simple  statements: 


WHY  ARITHMETIC  IS  TAUGHT  AT  PRESENT          41 

The  sum  of  the  same  powers  (above  the  second) 
of  two  integers  cannot  equal  a  perfect  power  of  the 
same  degree.  (In  the  case  of  the  second  degree 
there  are  any  number  of  examples,  as  32  +  42  =  52.) 
Fermat's  theorem. 

Every  even  number  is  the  sum  of  two  prime  num- 
bers. Goldbach's  theorem. 

The  consecutive  integers  8  and  9  are  exact 
powers;  are  there  any  other  consecutive  integers 
which  are  exact  powers  ?  Catalan. 


CHAPTER  III 

How  ARITHMETIC  HAS  DEVELOPED 

Reasons  for  studying  the  subject — The  historical  de- 
velopment of  the  reasons  for  teaching  arithmetic  has 
already  been  considered.  For  the  well-informed 
teacher  there  remain  two  other  historical  questions 
of  importance.  The  first  relates  to  the  development 
of  the  subject  itself,  and  the  second  to  the  methods  of 
teaching  it. 

There  are  good  and  sufficient  reasons  for  consider- 
ing briefly  the  history  of  arithmetic.  In  the  first 
place,  the  child  learns  somewhat  as  the  world  learns.1 
"The  individual  should  grow  his  own  mathematics, 
just  as  the  race  has  had  to  do.  But  I  do  not  propose 
that  he  should  grow  it  as  if  the  race  had  not  grown 
it  too.  When,  however,  we  set  before  him  math- 
ematics, —  be  it  high  or  low,  —  in  its  latest,  and  most 
generalized,  and  most  compacted  form,  we  are  trying 
to  manufacture  a  mathematician,  not  to  grow  one."2 
This  does  not  mean  that  the  child  must  go  through 

1  Cette  longue  education  de  1'humanite,  dont  le  point  de  depart  est  si 
loin  de  nous,  elle  recommence    en    chaque  petit    enfant.  —  Jean    Mace", 
L  'Arithmetique  du  Grand-Papa,  4^me  £d.,  p.  u. 

2  Jas.  Ward  in  the  Educational  Review,  Vol.  I,  p.  loo. 

42 


HOW  ARITHMETIC  HAS  DEVELOPED  43 

all  of  the  stages  of  mathematical  history  —  an  extreme 
of  the  "  culture-epoch "  theory ;  but  what  has  both- 
ered the  world  usually  bothers  the  child,  and  the 
way  in  which  the  world  has  overcome  its  difficulties 
is  suggestive  of  the  way  in  which  the  child  may  over- 
come similar  ones  in  his  own  development. 

In  the  second  place,  the  history  of  the  subject  gives 
us  a  point  of  view  from  which  we  can  see  with 
clearer  vision  the  relative  importance  of  the  various 
subjects,  what  is  obsolete  in  the  science,  and  what  the 
future  is  likely  to  demand.  Sterner1  has  compared 
the  teacher  of  to-day  to  a  traveller  who  by  much  toil 
has  reached  an  eminence  and  stops  to  take  breath  be- 
fore attempting  further  heights ;  he  looks  over  the  road 
by  which  he  has  journeyed  and  sees  how  he  might 
have  done  better  here,  and  made  a  short  cut  there, 
and  saved  himself  much  waste  of  time  and  energy 
yonder.  So  one  who  considers  the  historical  develop- 
ment of  arithmetic  and  its  teaching  will  see  how 
enormous  has  been  the  waste  of  time  and  energy, 
how  useless  has  been  much  of  the  journey,  and  how 
certain  chapters  have  crept  in  when  they  were  impor- 
tant and  remained  long  after  they  became  relatively 
useless.  He  will  see  the  subject  as  from  a  mountain 
instead  of  from  the  slough  of  despond  which  the  text- 
book often  presents,  and  he  will  be  able,  as  a  result, 
to  teach  with  clearer  vision,  to  emphasize  the  impor- 

1  Geschichte  der  Rechenkunst. 


44     THE  TEACHING  OF  ELEMENTARY  MATHEMATICS 

tant  and  to  minimize  or  exclude  the  obsolete,  and  thus 
to  save  the  strength  of  himself  and  of  his  pupils. 
He  will  also  learn  that  some  of  the  most  valuable 
parts  of  arithmetic  knocked  at  the  doors  of  the  schools 
long  centuries  before  they  were  admitted,  and  that 
teachers  have  had  to  struggle  long  and  persistently 
to  banish  some  of  the  most  objectionable  matter.  As 
a  result,  while  he  may  condemn  the  conservatism 
which  excludes  the  metric  system  and  logarithms  and 
certain  of  the  more  rational  methods  of  operations  to- 
day, he  will  have  more  faith  in  the  ultimate  success 
of  a  good  cause  and  will  see  more  clearly  his  duty 
as  to  its  advocacy. 

Extent  of  the  subject  —  It  is  manifestly  impossible 
to  give  more  than  a  glimpse  at  the  history  of  arith- 
metic. The  simple  question  of  numeration,  discussed 
with  any  fulness,  would  fill  a  volume  the  size  of  this 
one. 1  DeMorgan's  masterly  little  work,  "  Arithmetical 
Books,"  hardly  more  than  a  catalogue  (with  critical 
notes)  of  certain  important  arithmetics  in  his  library, 
fills  one  hundred  twenty-four  pages.2  For  the  stu- 
dent who  cares  to  enter  this  fascinating  field  some  sug- 
gestions are  given  in  a  subsequent  chapter.  But  for 
the  present  purpose  it  suffices  to  consider  merely  a 
few  important  events  in  the  general  development  of 
the  subject. 

1  See,  for  example,  Conant,  L.  L.,  The  Number  Concept,  New  York,  1896. 

2  London,  1847. 


HOW  ARITHMETIC  HAS  DEVELOPED  45 

The  first  step  —  counting  —  The  first  step  in  the  his- 
torical development  of  arithmetic  was  to  count  like 
things,  or  things  supposed  to  be  alike ;  in  the  broad 
sense  of  the  term  this  is  a  form  of  measurement.1 
Arithmetic  started  when  it  ceased  to  be  a  question  of 
this  group  of  savage  warriors  being  more  than  that, 
and  began  to  be  recognized  that  this  group  was  three 
and  that  two;  when  it  was  no  longer  a  matter  of  a 
stone  axe  being  worth  a  handful  of  arrow  heads,  but 
one  of  an  exchange  of  one  axe  for  eight  arrows. 
How  far  back  in  human  history  this  operation  goes 
it  is  impossible  to  say,  just  as  it  is  impossible  to  say 
how  far  back  human  history  itself  goes.  Indeed, 
counting  is  not  limited  to  the  human  family,  for 
ducks  count  their  young  and  crows  count  their  ene- 
mies.2 Any  discussion  of  the  nature  of  this  animal 
counting  must  lead  to  the  broader  question  of  the 
ability  to  think  without  words,  a  matter  so  foreign  to 
the  present  subject  as  to  have  no  place  here.8 

The  race  has  not,  however,  always  counted  as  at 
present.  It  was  a  long  struggle  to  know  numbers  up 

1  In  this  connection  the  teacher  should  read,  though  he  may  not  fully 
indorse,  Chap.  Ill  of  McLellan  and  Dewey's  Psychology  of  Number, 
New  York,  1895. 

2  This  subject  of  animal  counting  has  often  been  discussed.      It   is 
briefly  treated  in  the  chapter  on  Counting  in  Tylor's  Primitive  Culture, 
and  also  in  Conant's  Number  Concept  mentioned  on  p.  44. 

8  For  Max  M  tiller's  side  of  the  case  see  his  lecture  on  the  Simplicity  of 
Thought. 


46     THE  TEACHING  OF  ELEMENTARY  MATHEMATICS 

to  ten.  The  primitive  savage  counted  on  some  low 
scale,  as  that  of  two  or  three.  To  him  numbers  were 
"  i,  2,  many,"  or  "  i,  2,  3,  many,"  just  as  the  child 
often  says,  "  i,  2,  3,  4,  a  lot,"  and  somewhat  as  we 
count  up  very  far  and  then  talk  of  "  infinity." 

It  is  evident  that  there  must  be  some  systematic 
arrangement  of  numbers  in  order  that  the  mind  may 
hold  the  names.  For  example,  if  we  had  unrelated 
names  for  even  the  first  hundred  numbers,  it  would 
be  a  very  difficult  matter  to  teach  merely  their  se- 
quence, to  say  nothing  of  the  combinations.  But  by 
counting  to  ten,  and  then  (or  after  twelve)  combining 
the  smaller  numbers  with  ten,  as  in  three-ten  (thir- 
teen), four-ten  (fourteen),  .  .  .  twice-ten  (twenty),  and 
so  on,  the  number  system  and  the  combinations  are 
not  difficult. 

We  might  take  any  other  number  than  ten  for  the 
base  (radix).  If  we  took  three  we  should  count, 

one,  two,  three,  three-and-one, 

three-and-two,  two-threes,  .  .  .  , 

and  (with  our  present  numerals)  write  these, 
i,  2,  3,  ii  (i.e.,  one  three  and  one  unit),  12,  20,  ...  .* 

But  most  peoples,  as  soon  as  they  were  far  enough 
advanced  to  form  number  systems,  recognized  the 

1  A  brief  but  interesting  summary  of  this  subject  is  given  in  Fabrmann, 
K.  E.,  Das  rhythmische  Zahlen,  Plauen  i.  V,  1896,  p.  21.  It  is  also 
treated  in  numerous  text-books  and  elementary  manuals  in  English. 


HOW  ARITHMETIC  HAS  DEVELOPED  47 

natural  calculating  machine,  their  fingers,  and  hence 
began  to  count  on  the  scale  of  ten  (our  decimal 
system).  "  In  the  book  of  Problemata,  attributed  to 
Aristotle,  the  following  question  is  asked  (XV,  3): 
'  Why  do  all  men,  both  barbarians  and  Hellenes,  count 
up  to  10,  and  not  to  some  other  number?'  It  is 
suggested,  among  several  answers  of  great  absurdity, 
that  the  true  reason  may  be  that  all  men  have  ten 
fingers :  '  using  these,  then,  as  symbols  of  their  proper 
number  (viz.,  10),  they  count  everything  else  by  this 
scale.'"1 

To-day  it  is  common  to  hear  teachers  object  to 
allowing  a  child  to  count  on  his  fingers.  And  yet 
one  of  our  best  teachers  of  arithmetic  has  just  re- 
marked, what  is  indorsed  both  by  history  and  by  com- 
mon sense,  that  the  fingers  are  the  most  natural  and 
most  available  material.2  It  is  true  that  there  is  some 
ground  for  the  objection,  especially  on  the  part  of 
teachers  who  have  not  the  ability  to  lead  children 
to  rapid  oral  work;  but  if  the  world  had  not  counted 
in  this  way  we  should  not  have  had  our  decimal 
system. 

It  is  really  a  little  unfortunate,  arithmetically  con- 
sidered, that  man  has  ten  instead  of  twelve  fingers, 

1  Gow,  J.,  History  of  Greek  Mathematics,  Cambridge,  1884,  Chap.  I. 

2  Die  Finger  sind  also  das  naturlichste  und  nachste  Versinnlichungs- 
mittel.    Fitzga,  p.  82,  14,  59.     See  also  Conant's  Number  Concept,  p.  10, 
et  pass. 


48     THE  TEACHING  OF  ELEMENTARY  MATHEMATICS 

for  the  scale  of  twelve  is  the  easiest  of  all  the  scales. 
A  radix  must  not  be  too  small,  since  that  would 
require  too  much  labor  in  writing  comparatively  small 
numbers.  For  example,  on  the  scale  of  3,  fourteen 
would  appear  as  112  (i«32  -f  1-3  -f  2).  Neither  should 
the  radix  be  too  large,  since  there  must  be  ten  figures 
for  the  radix  ten,  twenty  for  the  radix  twenty,  and 
so  on,  and  too  many  characters  are  objectionable. 
Twelve,  like  ten,  is  a  medium  radix;  but  it  is  better 
than  ten  because  it  has  more  divisors.  Consider,  for 
instance,  the  fractions  most  commonly  used,  viz.,  J, 
^,  J,  ^.  These  are  written 

on  the  scale  of  10,     0.5,     0.333  .  .  .,     0.25,     0.125; 
on  the  scale  of  12,     0.6,     0.4,  0.3,      0.16. 

Hence  the  advantage  of  the  duodecimal  scale,  in  all 
work  involving  fractions,  is  apparent. 

Counting  must  have  preceded  notation  by  many 
generations,  just  as  talking  preceded  writing.  And 
while  there  are  good  reasons  for  teaching  the  num- 
erals to  a  child  while  he  is  learning  number  (the 
character  "3"  while  he  is  learning  to  pick  out  three 
things),  Pestalozzi  had  the  argument  of  race  develop- 
ment on  his  side  when  he  advocated  teaching  the 
characters  only  after  the  child  could  count  to  ten. 

And  in  teaching  the  child  number,  while  it  would 
be  very  logical  to  introduce  the  ratio  idea  first,  —  the 
idea  which  Newton  crystallized  in  his  well-known 


HOW  ARITHMETIC  HAS  DEVELOPED  49 

definition  of  number,  —  the  plan  is  not  in  harmony 
with  the  historical  development  of  the  race;  first, 
counting ;  second,  simple  operations ;  third,  a  notation ; 
this  is  the  race  order.  Aside  from  all  this,  there  is 
the  more  serious  question,  discussed  in  a  subsequent 
chapter,  as  to  the  psychological  phase  of  the  matter; 
whether  the  ratio  idea  is  not  altogether  too  abstract 
for  the  mind  of  the  child  beginning  to  study  num- 
ber. It  can  be  taught,  but  its  success  means  a  good 
teacher  with  a  poor  method,  a  David  with  a  sling. 
While  the  introduction  of  the  idea  in  the  beginning 
is  unwarranted  by  considerations  historical,  and  seems 
to  be  so  by  considerations  psychological,  it  is  desir- 
able as  soon  as  the  child  has  developed  sufficiently  to 
allow  it.  The  matter  has  not  yet  been  carefully 
enough  investigated,  however,  to  tell  just  when  this 
is.  Laisant,  who  does  not  lose  his  head  in  such 
affairs,  questions  whether  the  ratio  idea,  usually  rele- 
gated to  the  later  years  of  the  elementary  course, 
should  not  enter  very  early,  but  after  careful  con- 
sideration is  forced  to  the  conclusion  that  "number, 
in  its  elementary  form,  comes  to  us  by  the  evaluation 
of  collections  of  like  objects."1 

The  second  step  —  notation  —  Of  course  there  de- 
veloped in  connection  with  counting  a  certain  amount 
of  calculating  —  the  simplest  operations.  But  the 
second  step  of  great  importance  was  that  of  writing 

1  La  Mathematique,  p.  30,  31. 


50     THE  TEACHING  OF  ELEMENTARY  MATHEMATICS 

numbers.  The  plans  with  which  we  are  familiar,  the 
Hindu  ("Arabic")  and  the  Roman,  are  only  two  of 
many  which  have  been  used.  The  primitive  one  was 
that  of  simple  notches  in  a  stick  or  scratches  on  a 
stone.  But  of  scientific  systems  there  are  only  a  few 
types. 

The  Egyptians  had  a  system  much  like  the  Roman 
in  general  plan,  —  symbols  for  I,  10,  100  and  higher 
powers  of  ic.1 

The  Babylonians,  not  having  the  abundance  of 
stone  possessed  by  the  Egyptians,  resorted  to  writ- 
ing on  soft  bricks,  which  were  then  baked.  They 
therefore  developed  a  system  which  required  but  a 
few  characters  such  as  could  easily  be  impressed  by 
a  stick  upon  clay,  the  so-called  cuneiform  numerals. 
Their  symbols  were  three,  —  one  for  i,  one  for  10,  and 
one  for  ioo.2 

The  early  Greeks  used  the  initial  letters  of  the 
words  for  5,  10,  ioo,  1000,  10,000,  a  plan  leading  to 
a  system  about  like  the  Egyptian  and  Roman.  The 
late  Greeks  and  the  Hebrews  used  their  alphabets, 
giving  to  each  letter  a  number  value.  Thus  the 
Greeks  used  a  for  I,  fi  for  2,  7  for  3,  S  for  4,  e  for 
5,  an  old  form  called  digamma  for  6,  f  for  7,  rj  for 

1  Cantor  is,  of  course,  the  standard  authority  on  all  such  matters.     A 
good  summary  is  given  in  Sterner,  p.  17  seq. 

2  They  are  given  in  Beman  and  Smith's  translation  of  Fink's  History 
of  Mathematics,  Chicago,  1900. 


HOW  ARITHMETIC  HAS  DEVELOPED  51 

8,  and  0  for  9.  The  next  nine  letters,  with  one 
extra  symbol,  stood  for  tens,  i  —  10,  K  =  20,  X  =  30, 
etc.,  and  the  rest,  with  one  extra  character,  for  the 
hundreds.  The  system  was  a  difficult  one  to  master, 
but  it  enabled  the  computer  to  write  numbers  below 
1000  with  few  characters.  For  example,  387,  which 
the  Romans  wrote  CCCLXXXVII,  the  Greeks  wrote 
T7rr.i 

The  Romans  used  a  system  the  essential  features 
of  which  are  known  to  all.  The  origin  of  the  symbols 
has  long  been  a  matter  of  dispute,  but  they  are  now 
generally  recognized  to  be  modified  forms  of  old  Greek 
letters,  not  found  in  the  Latin  alphabet,  which  came 
through  the  Chalcidian  characters.2  The  Romans  in- 
troduced the  "  subtractive  principle "  of  writing  IV 
for  5—1,  XL  for  50  —  10,  etc.,  but  they  and  their 
successors  made  little  use  of  it.  The  tendency  to 
write  1 1 II  for  IV  is  still  seen  on  our  clock  faces.  The 
bar  over  a  number  was  rarely  used,  the  number  usually 
being  written  out  in  words  if  above  thousands,  while  the 
double  bar  sometimes  seen  in  American  examination 
questions,  and  the  idea  that  a  period  must  follow  a 
Roman  numeral,  may  be  called  stupid  excrescences  of 
the  nineteenth  century.  The  fact  that  the  Romans 

1  For  more  complete  discussion  see  Cantor,  I,  p.  117,  or  Sterner,  p.  50. 

2  Wordsworth,  Fragments  and  Specimens  of  the  Early  Latin,  p.  8; 
Fink's  History  of  Mathematics,  English,  by  Bcman  and  Smith,  p.   12 
Cantor,  I,  p.  486;    Sterner,  p.  78. 


52     THE  TEACHING  OF  ELEMENTARY   MATHEMATICS 

did  not  make  practical  use  of  their  system  in  writing 
large  numbers  should  show  us  the  criminal  waste  of 
time  in  requiring  children  of  our  day  to  bother  with 
the  system  beyond  thousands. 

The  Hindu  (or  so-called  Arabic)  system  can  be  traced 
back  to  certain  inscriptions  found  at  Nana  Ghat,  in 
the  Bombay  Presidency  (India),  and  first  made  known 
to  the  western  world  in  1877.  These  inscriptions 
probably  date  from  the  early  part  of  the  third  cen- 
tury B.C.  l  and  seem  to  prove  that  the  numerals  from 
4  to  9  inclusive  were  the  initial  letters  of  words  in 
the  ancient  Bactrian  alphabet.2  The  system  was  at 
that  time,  and  for  several  centuries  thereafter,  no 
better  than  many  others  of  antiquity,  because  it  had 
no  zero,  without  which  one  element  of  superiority, 
the  place-value  element,  is  wanting.  Without  the 
zero  we  cannot  write  ten,  one  hundred  six,  and  so 
on.  And  while  the  place  value  was  somewhat  ap- 
preciated as  early  as  the  time  of  the  cuneiform  nu- 
merals, the  zero  does  not  seem  to  have  appeared  in 
the  Hindu  system  before  300  A.D.,3  and  the  first 
known  use  of  the  symbol  in  a  document  dates  from 
four  centuries  later,  738  A.D.4 

There  is  much  question  as  to  the  way  in  which 
the  Hindu  numerals  first  entered  the  western  world. 

1  See  Journal  of  the  Royal  Asiatic  Society,  1882,  N.S.  XIV,  p.  336  ; 
1884,  N.S.  XVI,  p.  325  seq.,  especially  347. 

a  Cantor,  I,  p.  564.  8  Ib.,  p.  567.  *  Ib.,  p.  563. 


HOW  ARITHMETIC  HAS  DEVELOPED  53 

Sporadic  use  of  the  characters  is  found  before  the 
thirteenth  century.  But  about  1200  A.D.,  Leonardo 
Fibonacci,  of  Pisa,  returning  from  a  voyage  about  the 
Mediterranean,  brought  them  to  Italy.  Being  then 
in  use  in  various  Moorish  towns,  they  received  the 
name  "Arabic,"  although  the  Arabs  may  have  done 
nothing  more  than  to  disseminate  them  along  the 
borders  of  the  Occident.  If,  as  is  not  probable,1 
they  invented  the  zero,  they  deserve  to  have  the  name 
"Arabic"  continued,  but  if  not,  the  title  "Hindu  nu- 
merals "  is  much  to  be  preferred. 

It  was  nearly  a  century  later  than  Leonardo's  time 
before  the  system  had  penetrated  as  far  north  as 
Paris,2  and  it  was  not  until  about  1500  that,  thanks 
to  the  invention  of  printing,  it  began  to  get  a  firm 
footing  in  the  schools.8  For  teachers  who  await  with 
impatience  the  popular  use  of  the  metric  system,  or 
who  are  discouraged  by  the  apathy  of  their  co-workers 

1  Cantor,  I,  p.  569,  576. 

2  Henry,  Ch.,  Les  deux  plus  anciens  Traites  Frangais  d'Algorisme  et 
de  Geometric,   Boncompagni's  Bulletino,   February,    1882.     The   Ms.  is 
anonymous  and  was  written  about   1275  A.n. 

8  Those  who  are  interested  in  this  period  of  struggle,  from  1200  to  1500, 
will  find,  besides  the  discussions  in  Cantor,  Unger,  Sterner,  and  other 
writers  on  history,  some  interesting  facsimiles  in  Konnecke,  G.,  Bilder- 
atlas  zur  Geschichte  der  deutschen  Nationallitteratur,  Marburg,  1887, 
p.  40,  et  pass.  Halliwell,  J.  O.,  Kara  Mathematica,  London,  2d.  ed.,  1841, 
is  likewise  interesting  and  valuable,  as  is  also  the  pamphlet  edition  of 
"The  Crafts  of  Nombrynge,"  published  in  1894  by  The  Early  English 
Text  Society.  Boncompagni's  Bulletino  is,  of  course,  rich  in  material. 


54     THE  TEACHING  OF  ELEMENTARY  MATHEMATICS 

with  respect  to  the  use  of  logarithms  in  physical  com- 
putations, the  story  of  the  struggles  of  the  Hindu 
system  is  of  value. 

The  awkwardness  of  the  old  Roman  system,  in 
general  use  even  after  the  opening  of  the  sixteenth 
century,  is  well  seen  in  Kobel's  arithmetic,1  a  work 
which  barely  mentions  the  Hindu  numerals.  The 

following  is  a  specimen  :  "  If  you  would  add  —  to 
-  ,  write  them  crosswise  on  the  abacus;  then  by 

multiplying,  III  times  III  is  IX,  and  II  times  IV 
is  VIII;  add  the  VIII  and  IX  getting  XVII,  and 
this  is  the  numerator;  then  multiply  the  denomina- 
tors, III  times  IIII  is  XII;  write  the  XII  under 
the  XVII  and  make  a  little  line  between,  thus 

--—  —  ,   which   equals   one   and    ^rr."      Even   as  late 
XII 


as  1658,  when  Comenius  published  in  Niirnberg  the 
first  picture  book  for  the  instruction  of  children,  the 
well-known  Orbis  Pictus,  the  Roman  numerals  were 
in  common  use,  for  he  says,  "The  peasants  count 
by  crosses  and  half  crosses  (X  and  V)." 

The  next  great  step  in  arithmetic,  after  the  writing 
of  integers,  was  that  leading  to  a  knowledge  of  frac- 
tions. The  recognition  of  simple  fractions  is  pre- 
historic ;  but  the  struggle  to  compute  with  fractions 
extended  for  thousands  of  years  after  Ahmes  copied 

1  Das  new  Rechepiichlein,  1518,  quoted  here  from  Unger,  p.  16. 


HOW  ARITHMETIC  HAS  DEVELOPED  55 

his  famous  papyrus.  It  has  already  been  stated 
(p.  n)  that  the  ancient  Egyptians  could,  in  general, 
write  only  such  fractions  as  had  a  numerator  i,  and 
the  same  is  true  of  other  ancient  peoples.  The  later 
Greeks  wrote  the  numerator  followed  by  the  denomi- 
nator duplicated,  and  all  accented,  thus,  i?  tea"  /ca", 
for  J-J.1  The  Romans  had  a  fancy  for  fractions  with 
a  constant  denominator  as  a  power  of  12,  as  seen 
in  our  inch  (-^  of  a  foot),  and  the  Babylonians  for 
fractions  with  a  denominator  60  or  6O2,  as  seen  in  our 
minute  and  second  (i'  =  ^  of  a  degree,  i"  =  (^j)2  of 
a  degree). 

With  such  a  struggle  to  write  fractions,  it  is  not 
to  be  wondered  at  that  the  ancients  did  relatively 
little  in  arithmetical  computation,  or  that  the  child 
of  to-day  has  to  struggle  to  master  the  subject.  The 
world  could  solve  the  simple  equation  many  centuries 
before  it  could  do  much  with  fractions,  and  hence  it  is 
entirely  in  harmony  with  the  world  growth  to  introduce 
in  the  first  grade  such  simple  equations  as  2  +  (?)  =  7 
before  any  work  in  fractions  is  attempted. 

The  decimal  fraction  is  a  very  late  product  of 
arithmetical  ingenuity.  It  appeared  in  the  sixteenth 
century,  in  forms  like  •£$$$  and  5  ©  7  ®  8  ®,  for 
0.578,  and  about  1592  a  curve  was  used  by  Burgi 
to  cut  off  the  decimal  part.  But  in  1612,  Pitiscus 
actually  used  the  decimal  point,  and  the  system  was 
i  Cantor,  I,  p.  118. 


56     THE  TEACHING  OF  ELEMENTARY  MATHEMATICS 

perfected.1  It  was  not,  however,  until  well  into  the 
eighteenth  century  that  decimal  fractions  found  much 
footing  in  the  schools,  nor  was  it  until  the  nineteenth 
century  that  their  use  became  general.  During  the 
long  struggle  for  supremacy,  the  old-style  fraction 
was  literally  the  "common  fraction";  the  name  still 
survives,  although  the  decimal  form  is  now  by  far 
the  more  common. 

In  educational  circles  we  often  hear  advocated  the 
plan  of  teaching  decimal  fractions  before  common 
fractions.  But  to  attempt  any  theory  of  decimal  frac- 
tions first,  or  to  exclude  the  simplest  common  fractions 
from  the  first  year  of  arithmetic,  is  unscientific  from 
both  the  psychological  and  the  historical  standpoints. 
The  historical  order  is,  (i)  the  unit  fraction,  (2)  the 
common  fraction  (of  course  not  in  its  complete  de- 
velopment), and  (3)  the  decimal  fraction,  and  this  is 
also  the  natural  sequence  from  simple  to  complex, 
from  concrete  to  abstract. 

The  twofold  nature  of  ancient  arithmetic — As  has 
been  said,  arithmetic  was  studied  by  the  ancients  both 
as  a  utilitarian  and  a  culture  subject.  The  Greeks, 
for  example,  differentiated  the  science  into  Arithmetic 
(apiOwTi/cij)  and  Logistic  (Xo^icrrLKrj),  the  former  hav- 
ing to  do  with  the  theory  of  numbers,  and  the  latter 
with  the  art  of  calculating.2  Hence  when,  long  after, 

1  Cantor,  II,  p.  566-568. 

2  Gow,  J.,  History  of  Greek  Mathematics,  p.  22. 


HOW  ARITHMETIC  HAS  DEVELOPED  57 

these  two  branches  came  together  to  form  our  modern 
arithmetic,  the  subject  came  to  be  defined  as  "the 
science  of  numbers  and  the  art  of  computation,"  al- 
though the  modern  arithmetic  of  the  schools  includes 
much  besides  this. 

The  apiOprjTt/cij  of  the  Greeks  ran  also  into  the 
mystery  of  numbers,  and  much  was  made  of  this  sub- 
ject by  Pythagoras  (b.  about  580  B.C.)  and  his  fol- 
lowers. That  "there  is  luck  in  odd  numbers"  probably 
dates  back  to  his  school,  the  Latin  aphorism, 

"Deus  imparibus  numeris  gaudet," 
being  much  older  than  Virgil's  line, 

"Numero  deus  impare  gaudet."     (Eclogue  viii,  77.) 

The  mysticism  of  numbers,  the  universal  recognition 
of  3,  7,  and  9,  as  especially  significant,  forms  even 
now  an  interesting  study.  It  is  to  this  ancient  ten- 
dency that  we  owe  the  study,  only  recently  banished 
from  our  schools,  of  numbers  classified  as  amicable, 
deficient,  perfect,  redundant,  etc. 

The  art  of  calculating  (\oyuTTiicij)  among  the  ancients 
ran  largely  to  the  use  of  mechanical  devices,  such  as 
counters  (like  our  checkers),  and  the  abacus,  an  in- 
strument with  pebbles  (calculi,  whence  our  word  calcu- 
late) sliding  in  grooves  or  on  wires.  To-day  the 
Chinese  laundryman  in  America  still  performs  his 
calculations  on  an  abacus  (his  sivan  pan),  and  in 
Korea  the  school-boy  still  carries  to  school  his  bag  of 


58     THE  TEACHING  OF  ELEMENTARY   MATHEMATICS 

counters  (in  this  case  short  pieces  of  bone).  Among 
the  ancients,  too,  and  in  the  middle  ages,  finger- 
reckoning  was  a  recognized  part  of  the  necessary 
equipment  of  the  calculator.1 

It  is,  perhaps,  not  strange  that,  in  the  outburst  of 
enthusiasm  attendant  upon  the  introduction  of  the 
Hindu  numerals  in  the  schools  of  Western  Europe, 
these  mechanical  aids  should  have  been  relegated  to 
the  curiosity  shop.  Neither  is  it  strange  to  us,  looking 
back,  that  there  should  have  come  a  result  quite  un- 
foreseen by  the  educators  of  that  time,  namely,  a  loss 
of  the  power  of  real  insight  into  number.  Rules  for 
computation  existed  and  results  were  secured,  but  the 
realization  of  number  was  often  sadly  lacking.  It  was 
not  until  late  in  the  eighteenth  century  that  this  loss 
was  recognized  and  material  aids  to  a  comprehension 
of  number  were  restored  by  Busse,  Pestalozzi,  and 
their  associates. 

Arithmetic  of  the  middle  ages  —  Among  pre-Chris- 
tian Europeans  north  of  Italy  we  find  little  trace  of 
arithmetical  knowledge.  At  the  beginning  of  our  era 
learning  was  at  a  very  low  state  throughout  this  region. 
Tacitus  tells  us  that  writing  was  unknown  among  the 
common  people,  although  it  was  an  accomplishment  of 
the  priests.  As  business  increased,  however,  some 
mathematical  knowledge  became  necessary  even  before 
our  era.  Salt  and  amber  were  exported  from  Central 

1  For  description,  see  Gow,  p.  24. 


HOW  ARITHMETIC  HAS  DEVELOPED  59 

Europe,  and  Assyrian  inscriptions  tell  of  the  purchase 
of  the  latter  commodity  from  the  North.1  Tacitus  tells 
us  that  in  his  time  the  German  tribes  had  come  to 
know  the  Roman  weights  and  coins,  and  hence  they 
knew  enough  simple  counting  for  trading  purposes. 

To  replace  the  primitive  northern  arithmetic,  came, 
with  the  southern  conquerors,  the  Roman.  The  domi- 
nant power  soon  made  it  to  the  financial  interest  of 
the  traders  to  use  the  Italian  numerals.  And  although 
Rome  had  done  little  for  education,  some  of  her  later 
statesmen  recognized  the  value  of  scholarship,  as  wit- 
ness Capella,  Cassiodorus,  and  Boethius,  and  this  fact 
made  the  northern  tribes  incline  to  education.  Rome, 
however,  had  contributed  so  little  that,  when  her  power 
in  the  North  declined,  it  was  hardly  to  be  expected 
that  there  should  be  any  decided  contribution  to  knowl- 
edge among  her  former  subjects.  Nevertheless,  in 
Gaul,  where  the  Franks  established  a  well-ordered 
monarchy,  schools  were  founded,  and  the  French  king, 
Chilperic  (d.  584),  devoted  himself  with  earnestness 
to  a  system  of  public  education.  The  Merovingian 
princes  erected  a  kind  of  Court  school,  after  the  man- 
ner of  the  Romans,  and  thus  were  founded  the  Castle 
schools  which  were  common  throughout  the  middle 
ages.  Naturally,  however,  these  schools  contributed 
nothing  to  mathematics;  the  training  of  a  knight 
did  not  require  the  exact  sciences. 
1  Sterner,  p.  101. 


60     THE  TEACHING  OF  ELEMENTARY  MATHEMATICS 

The  Church  schools  did  more  for  mathematics,  as 
for  learning  in  general.  Wherever  the  Church  went, 
there  went  the  school.  By  whatever  name  known, 
whether  cloister,  cathedral,  or  parochial,  they  existed 
in  connection  with  every  large  ecclesiastical  founda- 
tion. Especially  did  the  schools  of  St.  Benedict  of 
Nursia,1  starting  from  the  parent  monastery  at  Monte 
Cassino  (near  Naples),  spread  all  over  Western  Europe, 
until  the  Benedictine  foundations  became  the  recog- 
nized centres  of  learning  from  the  Mediterranean  to 
the  North  Sea. 

In  these  Church  schools  mathematics  had  some  little 
standing.  The  quadrivium  of  arithmetic,  music,  ge- 
ometry, and  astronomy,  was  commonly  recognized  in 
higher  education,  and  in  spite  of  the  low  plane  on 
which  arithmetic  was  usually  placed  (see  p.  59),  some 
were  found  to  assign  it  a  worthy  place.2  To  Isidore, 
to  Bede  the  Venerable,  to  St.  Boniface,  to  Alcuin  of 
York,  and  other  Church  leaders,  we  owe  the  little 
standing  that  arithmetic  had  during  the  early  middle 
ages.  It  was  doubtless  at  Alcuin's  suggestion  that 
Charlemagne  decreed  that  the  schools  should  "  make 

1  480-543.     Called  by  Gregory  the  Great,  "scienter  nesciens,  et  sapi- 
enter  indoctus,"  learnedly  ignorant  and  wisely  unlearned. 

2  So  Isidore  of  Seville,  one  of  the  most  influential  of  mediaeval  writers, 
says:  "Tolle  numerum  rebus  omnibus  et  omnia  pereunt.     Adime  seculo 
computum  et  cuncta  ignorantia  caeca  complectitur,  nee  diflferi  potest  a 
ceteris    animalibus    qui    calculi    nescit    rationem."  —  Origines,   Lib.   Ill, 
cap.  4,  §  4. 


HOW  ARITHMETIC  HAS  DEVELOPED  6 1 

no  difference  between  the  sons  of  serfs  and  of  free 
men,  so  that  they  might  come  and  sit  on  the  same 
benches  to  study  grammar,  music,  and  arithmetic," l 
and  that  "the  ecclesiastics  should  know  enough  of 
arithmetic  and  astronomy  to  be  able  to  compute  the 
time  of  Church  festivals."2 

Brief  reference  has  already  (p.  5,  15)  been  made  to 
the  fact  that  men,  being  trained  in  the  monasteries 
for  ecclesiastical  work,  could  get  from  arithmetic  two 
things  which  correlated  with  their  professional  in- 
terests. One  was  the  ability  to  compute  the  date  of 
Easter  (whence  comes  the  chapter  on  the  calendar), 
and  the  other  was  the  training  in  disputation  and  in 
puzzling  an  opponent  (whence  come  many  inherited 
and  useless  puzzles  of  our  arithmetics  and  algebras 
of  to-day).  A  further  example  of  these  puzzles  of 
Alcuin's  time  may  be  of  interest:  "Two  men  bought 
some  swine  for  100  solidi,  at  the  rate  of  5  swine  for  2 
solidi.  They  divided  the  swine,  sold  them  at  the  same 
rate  at  which  they  bought  them,  and  yet  received  a 
profit.  How  could  that  happen?"3  The  puzzle  is 
unravelled  by  seeing  that  the  swine  were  of  different 
values.  There  were  120  sold  at  2  for  i  solidus,  120  at 
3  for  i  solidus,  so  that  5  went  for  2  solidi  as  before; 
120  good  ones  therefore  brought  60  solidi,  and  120 

1  Capitularies  of  789,  art.  70 ;  quoted  by  Guizot,  History  of  France,  I, 
p.  248. 

8  Sterner,  p.  no.  •  Cantor,  I,  p.  787  ;  Sterner,  p.  no. 


62     THE  TEACHING  OF  ELEMENTARY  MATHEMATICS 

poorer  ones  40  solidi,  so  the  dealers  had  their  100 
solid!  and  still  had  10  swine  left  by  way  of  profit. 

To  weed  out  problems  of  this  kind  has  taken  a 
long  time,  and  even  the  present  generation  finds  now 
and  then  some  advocate  of  exercises  almost  as  absurd, 
as  sharpeners  of  the  wit. 

The  period  from  Bede  to  the  tenth  century,  one 
of  the  darkest  of  the  middle  ages,  saw  arithmetic 
largely  given  up  to  the  computing  of  Easter,  the  com- 
putist  becoming  so  prominent  that  the  Germans  have 
designated  the  period  as  that  of  the  "  Computists."  1 

Another  movement  of  importance,  to  which  allusion 
has  already  been  made,  followed  this  period  of  degen- 
eracy. The  Hanseatic  League,  arising  from  a  union 
of  German  merchants  abroad  and  of  their  important 
commercial  centres  at  home,  attained  its  first  prom- 
inence in  the  thirteenth  century.  Although  it  had  for 
its  primary  object  the  protection  of  the  trade  routes 
between  the  allied  cities,  it  soon  developed  other  objects, 
such  as  the  assertion  of  town  independence  against  the 
rapacity  of  the  feudal  aristocracy,  the  establishment 
of  warehouses  along  the  paths  of  commerce,  the  formu- 
lation of  laws  of  trade,  and  the  general  improvement 
of  commercial  intercourse.  Among  these  acts  was  the 
establishment  of  the  Rechenschulen  (reckoning  schools, 
arithmetic  schools).  The  inadequacy  of  the  business 
course  in  the  Church  schools,  and  the  unsatisfactory 

1  Sterner,  p.  115  ;  but  see  Cantor,  I,  p.  783. 


HOW  ARITHMETIC  HAS  DEVELOPED  63 

attempts  at  teaching  bookkeeping,  arithmetic,  etc.,  led 
to  the  creation  of  the  office  of  Rechenmeister  already 
described.  The  guild  of  Rechenmeisters  included  some 
of  the  best  teachers  of  the  time,  —  Ulrich  Wagner  of 
Niirnberg,  who  wrote  the  first  German  arithmetic  (1482), 
Christoff  Rudolff,  who  wrote  the  first  German  algebra, 
Grammateus,  who  wrote  the  first  German  work  on  book- 
keeping, and  others  equally  celebrated.  So  powerful 
did  this  monopoly  become,  that  for  a  long  time  it  kept 
arithmetic  out  of  the  common  schools,  and  it  is  in  part 
due  to  this  influence  that  not  until  Pestalozzi's  time  was 
arithmetic  taught  to  children  on  entering  school. 

When  at  last  it  was  decided  that  arithmetic  could 
profitably  be  taught  in  the  earliest  grades,  the  inherited 
work  of  the  Rechenmeisters  was  dropped  in  upon  the 
lower  classes,  and  it  is  chiefly  due  to  this  fact  that  we 
have  had,  even  to  the  present  day,  a  mass  of  business 
problems  (often  representing  customs  of  the  days  of 
the  Rechenschulen,  but  long  since  obsolete,  like  part- 
nership involving  time)  in  the  fifth,  sixth,  and  seventh 
grades,  where  they  are  almost  wholly  unintelligible. 

The  period  of  the  Renaissance  —  The  period  of  the 
rebirth  of  learning,  the  Renaissance,  is  one  of  the  most 
interesting  which  the  historian  meets-  Manifold  causes 
contributed  to  make  the  close  of  the  fifteenth  century 
an  era  of  remarkable  mental  activity.  The  fall  of 
Constantinople  (1453)  turned  the  stream  of  Greek  cul- 
ture westward,  and  it  reached  the  shores  of  Italy  with 


64     THE  TEACHING  OF  ELEMENTARY  MATHEMATICS 

a  power  far  in  excess  of  that  which  it  exerted  in  the 
region  of  the  Bosphorus.  Joined  to  this  were  the 
revelations  of  that  new  astronomy  which,  by  the  help 
of  mathematics,  was  to  overthrow  the  Ptolemaic  theory ; 
the  discovery  of  a  new  continent  and  the  consequent 
revival  of  commerce ;  the  invention  of  cheap  paper  and 
of  movable  type,  two  influences  which  gave  wings  to 
thought ;  and,  not  the  least  of  all,  that  great  movement 
known  as  the  Reformation,  which  set  men  thinking  as 
well  as  believing.  From  this  period  of  migration,  of 
discovery,  of  invention,  and  of  independent  thought, 
dates  arithmetic  as  we  know  it. 

It  is  not  difficult  to  see  what  would  naturally  find 
place  in  arithmetic  at  that  time.  Crystallized  in  the 
new  printed  works  would  be  the  arithmetic  which  the 
Greeks  brought  from  Constantinople,  —  the  theory  of 
numbers  and  roots  by  geometric  diagrams.  The  Roman 
numerals,  which  had  been  used  almost  exclusively  to 
this  time,  would  find  a  prominent  place.  The  Arab 
arithmetic,  coming  in  with  the  Hindu  numerals  (already 
more  or  less  known),  would  contribute  its  little  share 
in  the  way  of  alligation,  Rule  of  Three  (our  simple 
proportion),  and  series,  which  last  was  known  in 
classical  times  as  well. 

Together  with  this  inherited  matter  would  naturally 
be  placed  the  arithmetic  demanded  by  the  peculiar 
conditions  of  the  time.  The  small  states,  with  their 
diverse  monetary  systems,  demanded  an  elaborate 


HOW  ARITHMETIC  HAS  DEVELOPED  65 

method  of  exchange,  not  merely  "simple,"  but  also 
"arbitrated."  The  absence  of  an  elaborate  banking 
system  like  that  of  to-day  rendered  the  common  draft 
one  payable  after,  instead  of  at  sight.  The  various 
systems  of  measures  in  the  different  states  and  cities 
required  elaborate  tables  of  denominate  numbers,1 
and  the  lack  of  decimal  fractions  explains  the  need 
of  compound  numbers  with  several  denominations. 
The  frequent  reductions  from  one  table  to  another, 
necessitated  by  these  circumstances,  encouraged  the 
use  of  the  Rule  of  Three  (Regula  de  tri,  Regeldetri, 
Regula  aurea),  so  that  this  piece  of  mechanism  came 
to  be  esteemed  quite  highly  in  the  arithmetics  of 
that  time.  Then,  too,  the  demands  of  commerce 
brought  in  problems  in  the  mensuration  of  masts  and 
sails,  and  those  which  finally  developed  in  our  Amer- 
ican text-books  as  General  Average.  Stock  com- 
panies not  having  as  yet  been  invented,  elaborate 
problems  in  partnership,  involving  different  periods 
of  time,  were  a  necessary  preparation  for  business. 
Later,  business  customs  demanded  Equation  of  Pay- 
ments, a  scheme  not  uncommon  in  days  when  long 
standing  accounts  were  the  fashion  between  whole- 
salers and  retailers.  Such  were  some  of  the  condi- 
tions in  the  days  when  printing  was  crystallizing  the 
science  of  arithmetic. 

JThus   Graffenried's  Arithmetica   Logistica,    1619,   has  21   pages  of 
tables. 


66     THE  TEACHING  OF  ELEMENTARY  MATHEMATICS 

Arithmetic  since  the  Renaissance  —  There  have  been 
several  improvements  in  methods  of  calculating  since 
the  period  of  revival  in  Italy,  and  the  business 
changes  have  revolutionized  the  commercial  side  of 
arithmetic. 

Among  the  improvements  in  pure  arithmetic,  the 
most  important  can  be  stated  briefly.  The  first  has 
to  do  with  the  invention  of  the  common  symbols  of 
operation,  which  may,  in  a  rough  way,  be  placed  in 
the  century  from  1550  to  I65O.1  Prior  to  this  time 
the  statement  of  the  operations  was  set  forth  in  full, 
and  for  any  material  advance  some  stenography  or 
symbolism  was  necessary. 

The  second  improvement  relates  to  the  invention 
of  decimal  fractions  about  1600,  an  invention  due 
perhaps  as  much  to  Biirgi  as  to  any  one.2  But  al- 
though these  fractions  appeared  three  centuries  ago, 
it  was  not  until  about  1750  that  they  found  much 
footing  in  the  schools,  so  conservative  are  schoolmas- 
ters, their  constituents,  and  the  various  examining 
authorities.  With  the  establishment  of  the  decimal 
fraction,  however,  arithmetic  was  revolutionized,  per- 
centage became  synonymous  with  advanced  business 
calculations,  the  greatest  common  divisor  (necessary 

1  A  brief  historical  note  upon  the  subject  may  be  found  in  Beman  and 
Smith's  Higher  Arithmetic,  Boston,  1896,  p.  43. 

2  Stevin,  Kepler,  Pitiscus,  and  others  had  a  hand  in  the  invention. 
See  Cantor,  II,  p.  567. 


^s* 

f        or  THE 
I  UNIVERSITY 

HOW  ARITHMETIC  HAS  DEVELOPED  67 

in  the  days  of  extensive  common  fractions)  became 
obsolete  for  scientific  purposes,  and  science  found  a 
new  servant  to  assist  in  her  vast  computations. 

The  third  improvement  is  the  invention  of  loga- 
rithms by  Napier  in  I6I4.1  One  might  expect  that  a 
scheme  which,  by  means  of  a  simple  table,  allowed 
computers  to  multiply  and  divide  by  mere  addition 
and  subtraction,  would  find  immediate  recognition  in 
the  schools.  And  yet,  so  conservative  is  the  pro- 
fession that,  even  in  high  schools  in  English  speak- 
ing countries,  logarithms  find  almost  no  place,  in 
spite  of  the  fact  that  neither  in  theory  nor  in  prac- 
tice do  they  present  any  difficulties  commensurate 
with  many  found  in  the  old-style  arithmetic.  In  Ger- 
many the  schools  are  more  progressive  in  this  matter. 

The  fourth  improvement  of  moment  is  seen  in  our 
modern  methods  of  multiplication  and  division.  A 
problem  in  division  three  hundred  years  ago  was  a 
serious  matter.  The  old  "scratch"  or  "galley" 
method2  was  cumbersome  at  the  best,  and  the  in- 
troduction of  the  "Italian  Method,"  which  we  com- 
monly use,  was  a  great  improvement.  Nor  is  the 
day  of  change  in  these  operations  altogether  passed, 


1  That  is,  his  "  Descriptio  mirifici  logarithmorum  canonis  "  appeared  in 
that  year.  The  best  brief  discussion  of  the  relative  claims  of  Napier  and 
Burgi  is  given  in  Cantor,  II,  p.  662  scq. 

a  Well  illustrated  in  Brooks,  E.,  Philosophy  of  Arithmetic,  Lancaster, 
Pa.,  1880,  p.  55,  59. 


68     THE  TEACHING  OF  ELEMENTARY   MATHEMATICS 

for  just  now  we  have  the  "Austrian  methods"  of 
subtraction  and  of  division  coming  to  the  front  in 
Germany,  and  we  may  hope  soon  to  see  them  com- 
monly used  in  the  English-speaking  world. 

The  fifth  improvement  is  partly  algebraic.  Algebra, 
as  we  know  it  with  its  present  common  symbolism, 
dates  only  from  the  early  part  of  the  seventeenth  cen- 
tury. With  its  establishment  there  departed  from  arith- 
metic all  reason  for  the  continuance  of  such  subjects  as 
alligation  (an  awkward  form  for  indeterminate  equa- 
tions), series  (better  treated  by  algebra),  roots  by  the 
Greek  geometric  process,  Rule  of  Three  (as  an  unex- 
plained rule),  and,  in  general,  the  necessity  for  any 
mere  mechanism.  Mathematicians  recognize  no  divid- 
ing line  between  school  arithmetic  and  school  algebra, 
and  the  simple  equation,  in  algebraic  form,  throws  such 
a  flood  of  light  into  arithmetic  that  hardly  any  leading 
educator  would  now  see  the  two  separated. 

The  present  status  of  school  arithmetic  is  one  of 
unrest.  We  have  these  inheritances  from  the  Renais- 
sance, and  with  difficulty  we  are  breaking  away  from 
them.  Only  recently  have  we  seen  alligation  disap- 
pear from  our  text-books,  and  slowly  but  surely  are 
we  driving  out  "true"  discount,  equation  of  payments, 
arbitrated  exchange,  troy  and  apothecaries'  measures, 
compound  proportion,  and  other  objectionable  matter. 
Such  subjects,  are,  as  already  suggested,  unworthy  of 
a  place  in  the  course  which  is  to  fit  for  general  citi- 


HOW  ARITHMETIC   HAS   DEVELOPED  69 

zenship;  for  they  are  practically  obsolete  (like  troy 
weight),  or  useless  (like  arbitrated  exchange),  or  mere 
mechanism  and  show  of  knowledge  (like  compound 
proportion),  or  they  give  a  false  idea  of  business  (like 
"true"  discount). 

Slowly  we  are  opening  the  door  to  the  simple  equa- 
tion, because  it  illuminates  the  practical  problems  of 
arithmetic,  especially  those  of  percentage  and  propor- 
tion. "  It  is  evident,"  says  M.  Laisant,  "  that  all 
through  the  course  of  arithmetic,  letters  should  be  \C^ 
introduced  whenever  their  use  facilitates  the  reasoning 
or  search  for  solutions."1 

The  present  tendency  is  decidedly  in  favor  of  elimi- 
nating the  obsolete,  of  substituting  modern  business  for 
the  ancient,  of  destroying  the  artificial  barrier  between 
arithmetic  and  algebra,  and  of  shortening  the  course  in 
applied  arithmetic.  As  the  report  of  the  "Committee 
of  Ten  "  stated  the  case,  "  The  conference  recommends 
that  the  course  in  arithmetic  be  at  the  same  time 
abridged  and  enriched;  abridged  by  omitting  entirely 
those  subjects  which  perplex  and  exhaust  the  pupil 
without  affording  any  really  valuable  mental  discipline, 
and  enriched  by  a  greater  number  of  exercises  in  simple 
calculation  and  in  the  solution  of  concrete  problems."2 
Three  years  later,  the  "  Committee  of  Fifteen  "  had  this 

1  La  Mathematiquc,  p.  206. 

2  For  full  report  of  the  mathematical  conference,  see  Bulletin  No.  205, 
United  States  Bureau  of  Education,  Washington,  1893,  p.  104. 


70     THE  TEACHING  OF  ELEMENTARY   MATHEMATICS 

further  suggestion :  "  Your  Committee  believes  that, 
with  the  right  methods,  and  a  wise  use  of  time  in  pre- 
paring the  arithmetic  lesson  in  and  out  of  school,  five 
years  are  sufficient  for  the  study  of  mere  arithmetic  — 
the  five  years  beginning  with  the  second  school  year 
and  ending  with  the  close  of  the  sixth  year ;  and  that 
the  seventh  and  eighth  years  should  be  given  to  the 
algebraic  method  of  dealing  with  those  problems  that 
involve  difficulties  in  the  transformation  of  quantitative 
indirect  functions  into  numerical  or  direct  quantitative 
data." l 

In  all  this  present  change  and  suggestion  of  change, 
the  radical  element  in  the  profession  is  restrained  by 
several  forces :  the  publisher  fears  to  join  in  a  too 
pronounced  departure;  the  author  is  also  concerned 
with  the  financial  result ;  the  teacher  is  fearful  of  the 
failure  of  his  pupils  on  some  official  examination  (a 
most  powerful  influence  in  hindering  progress);  and 
the  pupil  and  his  parents  see  terrors  in  any  depart- 
ure from  established  traditions.  But  in  spite  of  all 
this,  the  improvement  in  the  arithmetics  in  America 
has,  within  a  few  years,  been  very  marked  —  more  so 
than  in  any  other  country. 

1  Report  of  the  Committee  of  Fifteen,  Boston,  1895,  P*  24' 


CHAPTER   IV 
How  ARITHMETIC  HAS  BEEN  TAUGHT 

The  value  of  the  investigation  of  the  way  in  which 
arithmetic  has  been  taught,  especially  during  the  nine- 
teenth century,  is  apparent.  Find  the  methods  fol- 
lowed by  the  most  successful  teachers,  find  the  failures 
made  by  those  who  have  experimented  on  new  lines, 
and  the  broad  question  of  method  is  largely  settled. 
"  The  science  of  education  without  the  history  of  educa- 
tion is  like  a  house  without  a  foundation.  The  his- 
tory of  education  is  itself  the  most  complete  and 
scientific  of  all  systems  of  education."1 

It  is  impossible  at  this  time  to  trace  the  develop- 
ment of  the  general  methods  of  teaching  the  subject, 
up  to  the  opening  of  the  nineteenth  century.  Already, 
in  Chapter  I,  the  development  of  the  reasons  for 
teaching  the  subject  has  been  outlined,  and  from  this 
the  general  methods  employed  may  be  inferred. 
Only  a  hurried  glance  at  a  few  of  the  more  interest- 
ing details  is  possible. 

The  departure  from  object  teaching  —  Arithmetic, 
at  least  in  the  Western  world,  was  always  based  upon 
object  teaching  until  about  1500,  when  the  Hindu 

1  Schmidt,  Geschichte  der  Padagogik,  I,  p.  9. 
7' 


L^ 


P 


72     THE  TEACHING   OF   ELEMENTARY   MATHEMATICS 

numerals  came  into  general  use.  But  in  the  enthu- 
siasm of  the  first  use  of  these  symbols,  the  Christian 
schools  threw  away  their  abacus  and  their  numerical 
counters,  and  launched  out  into  the  use  of  Hindu 
figures.  And  while  they  saw  that  the  old-style  ob- 
jective work  was  unnecessary  for  calculation,  which 
is  true,  they  did  not  see  that  it  was  essential  as  a 
basis  for  the  comprehension  of  number  and  for  the 
development  of  the  elementary  tables  of  operation. 
Hence  it  came  to  pass  that  a  praiseworthy  revolution 
in  arithmetic  brought  with  it  a  blameworthy  method 
of  teaching.  Although  there  were  better  tools  for 
work  —  the  Hindu  numerals,  arithmetic  became  even 
more  mechanical  than  before,  and  it  was  not  until  the 
time  of  Pestalozzi,  three  centuries  later,  that  educators 
awoke  to  the  great  mistake  which  had  been  made  in 
discarding  objects  as  a  basis  for  number  teaching. 

With  the  introduction  of  the  Eastern  figures,  text- 
books became  filled  with  rules  for  operations,  and 
teachers  followed  books  in  this  mechanical  tendency. 
To  define  the  terms,  to  learn  the  rules,  to  repeat  the 
book,  this  was  the  almost  universal  method  for  three 
hundred  years  before  Pestalozzi,  and  even  yet  the 
method  has  not  entirely  died  out.1  A  modern  math- 

1  Janicke  and  Schurig's  Geschichte  der  Mcthodik  des  Unterrichts  in  den 
mathematischen  Lehrfachern,  Band  III  of  Kehr's  Geschichte  der  Meth- 
odik  des  deutschen  Volksschulunterrichtes,  Gotha,  1888.  The  first  part 
of  the  volume  is  Janicke's  Geschichte  der  Methodik  des  Rechenun- 
terrichts,  and  will  hereafter  be  referred  to  as  Janicke.  Janicke,  p.  21. 


HOW  ARITHMETIC  HAS  BEEN  TAUGHT  73 

ematician  would  fare  ill  in  passing  an  arithmetic  ex- 
amination of  those  days,  before  their  examiners,1 
just  as  the  mathematician  of  a  couple  of  centuries 
hence  will  wonder  at  the  absurdities  of  many  of  o\ir 
questions  to-day. 

Rhyming  arithmetics  —  The  difficulty  of  committing 
to  memory  a  large  number  of  rules  upon  the  subject 
led  educators  to  look  for  a  remedy.  Some,  and 
among  them  Ascham  and  Locke,  mildly  protested 
against  so  many  rules,  but  for  a  long  time  a  large 
number  was  considered  necessary,  and  this  plan  is 
even  yet  advocated  by  many  teachers.  Among  the 
remedies  suggested  was  that  of  putting  the  rules  in 
rhyme,  the  argument  being  that  ( i )  a  multitude  of  rules 
is  a  necessity,  (2)  rhymes  are  easily  memorized,  (3) 
hence  this  multitude  of  rules  should  appear  in  rhyme, 
—  a  good  enough  syllogism  if  we  admit  tjje,  major 
premise.  Hence  for  a  long  time  rhyming  Ales  were  in 
vogue,  and  might  be  to-day  had  not  opinions  changed 
as  to  the  value  of  the  rule  itself.  Even  during  the 
last  quarter  of  the  nineteenth  century,  however,  an 
arithmetic  in  rhyme  appeared  in  New  York  State  — 
so  little  are  the  lessons  of  the  history  of  methods 
known. 

Form  instead  of  substance  was  a  natural  outcome 
of  the  policy  of  making  arithmetic  purely  mechanical. 
So  we  find  much  attention  paid  to  the  preparation  of 

>  For  such  a  paper  see  Janicke,  p.  22. 


74     THE  TEACHING  OF  ELEMENTARY  MATHEMATICS 

artistic  copybooks  with  curious  arrangements  of  work. 
The  following  may  serve  to  illustrate  the  results  of 
this  tendency:1 

79745  97548 

64789  69457 


5160119905  6775391436 

It  is  possible  that  to  this  tendency  to  prepare  artis- 
tic copybooks  rather  than  to  acquire  facility  in  arith- 
metic there  is  to  be  attributed  the  continued  use  of 
the  old  "scratch"  or  "galley"  method  of  division,2 
long  after  the  more  modern  Italian  method  was 
known. 

Instruction  in  method,  for  teachers  of  arithmetic, 
began  to  appear  in  noteworthy  form  about  the  mid- 
dle of  the  seventeenth  century,  "like  an  oasis  in  a 

1  Janicke,  p.  27. 

2  This  method  is  given  in  all  of  the  histories  of  mathematics  already 
named. 


HOW  ARITHMETIC   HAS  BEEN  TAUGHT  75 

desert,"  says  Janicke.  But  the  plans  suggested  were 
still  mechanical  —  counting  and  writing  numbers  in 
unlimited  number  space,  then  addition  in  such  space, 
then  subtraction,  and  so  on.  "The  teacher,"  says 
one  of  the  best  works  of  the  time,  "is  to  write  the 
first  nine  numbers,  then  pronounce  them  four  or  five 
times,  then  let  the  boys,  one  after  another,  repeat 
them." 

A  picture  of  the  best  methods  employed  at  the 
opening  of  the  eighteenth  century  may  be  seen  in 
the  rules  for  the  celebrated  Francke  Institute  at 
Halle  (i/oa),1  rules  not  without  suggestiveness  to  cer- 
tain teachers  to-day : 

"All  children  who  can  read  shall  study  arithme- 
tic." It  was  not  until  about  a  century  later  that  the 
subject  was  taught  to  children  just  entering  school, 
and  to-day  we  have  quite  a  pre-Pestalozzian  move- 
ment to  the  old  plan,  akin  to  pre-Raphaelitism  in  the 
graphic  arts. 

"On  account  of  the  diverse  aptitudes  of  children, 
in  the  matter  of  arithmetic,  it  is  impossible  to  form 
classes;  hence  the  teacher  shall  use  a  printed  book 
and  shall  t£ach  the  subject  from  it.  ...  He  shall 
go  around  among  the  children  and  give  help  where 

1  Unger,  p.  140  ;  Janicke,  p.  32.  In  general  it  may  be  said  that  any 
one  who  wishes  to  follow  the  development  of  method  in  arithmetic  must 
consult  these  works.  There  is  nothing  more  systematic  than  Unger, 
nothing  so  complete  as  Janicke. 


76     THE  TEACHING  OF   ELEMENTARY   MATHEMATICS 

it  is  necessary."  To-day  we  hear  not  a  little  of  "the 
laboratory  method"  and  "individual  teaching,"  a  re- 
turn to  the  methods  of  the  past,  methods  in  which 
the  inspiration  of  community  work  was  wanting, 
methods  long  since  weighed  in  the  balance  and 
found  wanting. 

"The  teacher  must  dictate  no  examples,  but  each 
child  shall  copy  the  problems  from  the  book  and 
work  them  out  in  silence."  This  plan  is  also  not 
unknown  in  the  teaching  of  the  subject  to-day. 

"  It  would  be  a  good  thing  if  the  teacher  would 
himself  work  through  (durchrechnet)  the  book  so  that 
he  could  help  the  children  " ! 

It  was  toward  the  close  of  the  eighteenth  century 
that  the  modern  treatment  of  elementary  arithmetic 
began  to  show  itself.  In  the  Philanthropin  at  Dessau, 
an  institution  to  which  education  owes  not  a  little, 
we  find  in  1776  very  little  improvement  upon  the  old 
plan  of  pretending  to  teach  all  of  counting,  then  all 
of  addition,  then  all  of  subtraction,  and  so  on.1  But 
in  the  following  year  Christian  Trapp  began  upon 
entirely  new  lines,  and  in  1780  he  published  his  "Ver- 
such  einer  Padagogik,"  in  which  he  worked  out  quite 
a  scheme  of  teaching  young  children  how  to  add  and 
subtract,  objects  being  employed  and  the  effort  being 
made  to  teach  numbers  rather  than  figures.  This  he 
followed  by  simple  work  in  multiplication  and  division, 

1  Janicke,  p.  44. 


HOW  ARITHMETIC  HAS  BEEN  TAUGHT  77 

and  he  worked  out  a  systematic  use  of  a  box  of  blocks 
illustrating  the  relation  of  tens  to  units,  a  forerunner 
of  the  Tillich  reckoning-chest  mentioned  later.1  It  is 
here  that  we  may  say,  with  fair  approximation  to  justice, 
the  modern  teaching  of  elementary  arithmetic  begins. 

Trapp's  successor  was  Gottlieb  von  Busse,  whose  first 
works  on  arithmetic  appeared  in  1786.      He  was  still 
wedded   to   the  old  system  of   first  teaching  numera- 
tion (to  trillions),  then  the  four  fundamental  processes 
in  order,  and  so  on.     But  at  the  same  time  he  made 
a  distinct  advance  in  the  systematic  use  of    •    •    • 
number  pictures  (Zahlenbilder,  translated  by 
some  genius  as  "number  builders"  !\  points 


being  associated  with  the  group  as  here 
shown.  He  used  special  forms  for  tens  (to  dis- 
tinguish them  from  the  unit  dots),  and  also  for  the 
hundreds  and  the  thousands,  thus  carrying  a  good 
thing  to  a  ridiculous  extreme.2  In  the  same  way  we 
still  have  in  our  day  not  a  few  failures  as  a  result 
of  carrying  objective  teaching  too  far.  This  is  one 
of  Grube's  errors,  although  few  would  follow  him 
closely  enough  to  be  harmed  by  it. 

Mention  should  also  be  made  of  the  work  of  a 
nobleman,  Freiherr  von  Rochow,  of  Rekan,  near 
Brandenburg,  who  is  known  as  the  reformer  of  the 
country  schools  of  Germany,8  and  whose  influence  led 

1  Janicke,  p.  44.  2  Ib.,  p.  45  seq.  ;  Unger,  p.  165  scq. 

8  Unger,  p.  138. 


78     THE  TEACHING  OF  ELEMENTARY  MATHEMATICS 

to  the  attempt  on  the  part  of  his  assistants  to  make 
arithmetic  attractive  instead  of  insufferably  dull,  and 
to  use  it  for  training  the  mind  as  well  as  for  a  prepa- 
ration for  trade.1 

Pestalozzi  —  Trapp,  Busse,  von  Rochow,  and  a  few 
others  whose  names  and  work  can  hardly  be  men- 
tioned here,  were  like  "the  voice  of  one  crying  in 
the  wilderness  " ;  there  was  another  who  should  come. 
Johann  Heinrich  Pestalozzi,  a  poor  Swiss  schoolmaster, 
a  man  who  seemed  to  make  a  failure  of  whatever  he 
undertook,  laid  the  real  foundation  of  primary  arith- 
metic as  it  has  since  been  recognized.  He  wrote  no 
work  directly  upon  the  subject,  and  one  who  searches 
for  his  ideas  upon  number  teaching  has  to  pick  a 
little  here  and  a  little  there  from  among  his  numerous 
papers  and  letters,  and  take  the  testimony  of  those 
who  knew  him.2 

Number  had  been  taught  to  children  by  the  aid 
of  objects  before  Pestalozzi  began  his  work.  This, 
indeed,  as  already  stated,  was  the  primitive  plan,  and 
was  thrown  over  only  with  the  introduction  of  print- 
ing and  the  Hindu  numerals.  Trapp  and  Busse  had 
tried,  not  to  revive  the  old  plan  of  using  objects 
for  all  calculations,  but  to  make  a  reasonable  use  of 
objects  with  beginners.  Their  plans  were  crude, 
however,  and  it  was  reserved  for  Pestalozzi  scientif- 

1  Janicke,  p.  48,  46.  2  Ib.,  p.  63  j  Unger,  p.  176. 


HOW  ARITHMETIC  HAS  BEEN  TAUGHT  79 

ically  to  make  perception  the  basis  for  all  number 
work.1 

Of  course  this  does  not  mean  that  Pestalozzi  was 
the  first  to  recognize  the  value  of  perception.  This 
was  not  at  all  new.  The  ancients  understood  it 
well,  and  Horace  even  placed  it  in  his  verse:  "The 
things  which  enter  by  the  ear  affect  the  mind  more 
languidly  than  such  as  are  submitted  to  the  faithful 
eyes."  2 

Pestalozzi,  however,  was  the  first  to  recognize  its 
value  to  the  full,  and  to  put  it  to  practical  use  in 
teaching.8 

With  Pestalozzi,  too,  the  formal  culture  value  of 
number  came  definitely  and  systematically  to  the  front, 
the  value  of  "mental  gymnastic"  (Geistesgymnastik) 
was  recognized  —  unduly  so,  to  be  sure,  and  all  daw- 
dling "  busy  work  "  was  wanting.  The  children  worked 
rapidly,  cheerfully,  orally.  They  showed  themselves 
quick  in  number  work,  wide  awake,  active,  and  we  can 
learn  more  to-day  from  Pestalozzi  than  from  any  other 
one  teacher  of  the  subject,  and  this  in  spite  of  all  the 
faults  of  method  which  he  unquestionably  possessed. 

1  "  Die  Anschauung  ist  das  absolute  Fundament  aller  Erkenntniss." 
—  Pestalozzi  to  Gessner.    Compare  Diesterweg :  "  Das  ganze  Geheimniss 
der  Elementarmethode  ruht  in  der  Anschaulichkeit." 

2  "  Segnius  irritant  animos  dimissa  per  aurem, 

Quam  quae  sunt  oculis  subiecta  fidelibus."  —  Ars  poetica,  v.  180. 
8  Shafer,    Fr.,    Geschichte    des    Anschauungsunterrichts,    in    Kchr's 
Geschichte  der  Methodik,  I,  p.  468. 


80     THE  TEACHING  OF  ELEMENTARY  MATHEMATICS 

It  is  related  of  him1  that  a  Niirnberg  merchant,  who 
had  heard  with  some  doubts  of  his  success  in  teaching 
arithmetic,  came  to  the  school  one  day  and  asked  to 
be  allowed  to  question  the  boys.  The  request  being 
granted,  he  proposed  a  rather  complicated  business 
problem  involving  fractions.  To  his  astonishment  the 
boys  inquired  whether  he  wished  it  solved  in  writing 
or  "in  the  head/'  and  upon  his  naming  the  latter  plan 
he  began  for  himself  to  figure  out  the  result  on  paper ; 
but  before  he  had  half  done  the  boys'  answers  began 
to  come  in,  so  that  he  left  with  the  remark,  "  I  have 
three  youngsters  at  home,  and  each  one  shall  come 
to  you  as  soon  as  I  can  get  there."2  The  incident, 
possibly  exaggerated,  is  not  unique ;  Biber 3  and  others 
relate  numerous  instances  of  the  success  which  at- 
tended Pestalozzi's  earnest  work  in  oral  arithmetic 
founded  upon  perception. 

Pestalozzi  was  not  narrow  in  his  ideas  as  to  the 
objects  to  be  employed,  as  Tillich  and  many  other 
teachers  of  later  times  have  been.  This  particular 
device  (say  some  form  of  abacus),  or  that  (as  some 
set  of  cubes,  or  disks,  or  other  geometric  forms),  did 
not  appeal  to  him.  He  used,  to  be  sure,  an  arrange- 


1  By  Blockmann,  "  Heinrich  Pestalozzi,  Ziige  aus  dem  Bilde  seines 
Lebens,"  Dresden,  1846. 

2  See  also  De  Guimp's  Pestalozzi,  American  ed.,  p.  214. 

3  Life   of  Pestalozzi,  p.  227  et  pass.     It  is  unfortunate  that  this  ex- 
cellent work  has  become  so  rare. 


HOW  ARITHMETIC  HAS  BEEN  TAUGHT  8 1 

ment  of  marks  on  a  chart  (his  "  units'  table,"  Einheits- 
tabelle),  but  he  did  not  limit  himself  to  any  such 
device;  he  led  the  child  to  consider  all  objects  which 
were  of  interest  to  him,  nor  did  he  fear  (O  modern 
teacher!)  to  let  him  use  the  most  natural  calculating 
device  of  all  —  the  finger^.1 

Pestalozzi's  leading  contributions  may  be  summed  up 
as  follows: 

1.  He  taught  arithmetic  to  children  when  they  firsL 
came  to  school,  basing  his  work  upon  perception,  and 
seeking  to  make  the  child  independent  of  all  rules 
and  traditions.     Nevertheless,  he  did  not  wholly  free 
the  subject  from  mechanism.     He  avoided  the  baser 
form  which  depended  upon  rules  and  principles,  but 
he  substituted  a  mechanism  of  forms  based  upon  per- 
ception.     His  never  ending   2x1  +  3x1=  ?xi    is 
very  tiresome  in  spite  of  its  value  for  beginners.2 

2.  He  insisted  that  the  knowledge  of  number  should 
precede  the  knowledge  of  figures  (Hindu  numerals),  in 
the  number  space  from  i  to  10.    "  Now  it  is,"  said  he, 
"a  matter  of  great  importance  that  this  ultimate  basis 
of  all  number  should  not  be  obscured  in  the  mind  by 

1  The  best  insight  into  Pestalozzi's  ideas  along  this  line  is  given  in 
the  work  of  his   friend   and   co-worker,  Krusi,  Anschauungslehre  der 
Zahlenverhaltnisse,  Zurich,  1803. 

2  "  Damit  fuhrte  er  in  der  Darbietung  vom  vorpestalozzischen  puren 
Mechanismus   zum   anschaulichen   Zahlmechanismus,  an  dem  unser  ele- 
mentarer    Rechenunterricht    auch    heute    noch     krankt."       Brautigam, 
Mcthodik  des  Rechen-Unterrichts,  2.  Aufl.,  Wien,  1895,  P-   2- 

a 


82     THE  TEACHING  OF  ELEMENTARY  MATHEMATICS 

arithmetical  abbreviations." l  Tillich,  Pestalozzi's  most 
talented  follower,  agrees  with  his  master  in  this.  "  The 
figures,"  he  writes,  "are  only  the  symbols  for  numbers. 
Hence  they  ought  not  to  be  taught  to  the  child  until 
the  numbers  are  familiar  to  him.  To  do  otherwise  is 
to  make  the  same  mistake  that  one  would  make  in 
teaching  letters  to  a  child  who  could  not  yet  talk,"2  a 
rather  radical  statement,  but  one  with  a  core  of  truth. 
First  and  foremost  the  child  must  conceive  of  number; 
figures,  operations,  applications  beyond  mere  counting 
and  selecting  of  groups,  these  could  wait.  As  one  of 
the  modern  opponents  of  Grube's  heresy  has  put  it, 
"  First  the  number  concept,  then  the  operations." 3 

3.  He   also   insisted    that    the   child    should    know 
the   elementary  operations   before  he  was  taught  the 
Hindu  numerals.     "When  a  child  has  been  exercised 
in  this  intuitive  method  of  calculation  as  far  as  these 
tables  go  (i.e.  from  i  to    10),    he  will   have   acquired 
so  complete  a  knowledge  of   the  real  properties  and 
proportions   of   number   as  will   enable   him   to   enter 
with   the   utmost  facility  upon   the  common   abridged 
methods  of  calculating  by  the  help  of  ciphers."4 

4.  The   Hindu   numerals   followed   this   training   in 
pure    number.      "  His  mind  is  above  confusion  and 

1  Letter  to  Gessner,  Biber's  Pestalozzi,  p.  278. 

2  Lehrbuch  der  Arithmetik,  p.  41. 

8  Beetz,  K.  O.,  Das  Wesen  der  Zahl,  p.  204. 
*  Letter  to  Gessner,  Biber's  Pestalozzi,  p.  282. 


HOW  ARITHMETIC  HAS  BEEN  TAUGHT  83 

trifling  guesswork;  his  arithmetic  is  a  rational  pro- 
cess, not  a  mere  memory  work  or  mechanical  routine ; 
it  is  the  result  of  a  distinct  and  intuitive  apprehen- 
sion of  number." 1 

5.  Fractions  were  treated  in   the  same  way;  first 
the  concept  of  fraction,  then  some  exercise  in  opera- 
tions,  finally   the    shorthand    characters.      After    the 
child   has  "such  an   intuitive  knowledge   of  the  real 
proportions    of    the    different    fractions,   it  is   a  very 
easy  task  to  introduce  him  to  the  use  of  ciphers  for 
fraction   work."2      After    all,    Pestalozzi    was    simply 
following    out     RatkeV -well-known    rule,    "  First    a 
thing  in  itself,  arilTthen.  the  way  of  it;  matter  before 
form."     The  only  question   is,    Did   he   postpone  the 
form  too  long? 

6.  He  made  arithmetic  the   most  prominent  study 
in  the  curriculum.     "Sound  and   form  often  and  in 
various  ways   bear    the   seeds  of    error    and    deceit; 
number   never;   it   alone    leads   to   positive    results."8 
"I  made  the  remark,"  said  Pere  Girard,  himself  one 
of  the  foremost   Swiss   educators,  "to   my  old  friend 
Pestalozzi,  that  the  mathematics  exercised  an  unjusti- 
fiable sway  in   his  establishment,  and  that   I   feared 
the  results   of  this  on  the  education  that  was  given. 
Whereupon   he  replied  to  me  with  spirit,  as  was  his 
manner,    'This    is    because    I   wish    my  children    to 

1  Letter  to  Gessner,  Biber's  Pestalozzi,  p.  282.  *  Ib.,  p.  283. 

8  Pestalozzi's  Sammtliche  Werken,  n.  Bd.,  p.  226. 


84     THE  TEACHING  OF  ELEMENTARY  MATHEMATICS 

believe  nothing  which  cannot  be  demonstrated  as 
clearly  to  them  as  that  two  and  two  make  four.' 
My  reply  was  in  the  same  strain :  '  In  that  case,  if 
I  had  thirty  sons,  I  would  not  intrust  one  of  them 
to  you,  for  it  would  be  impossible  for  you  to  dem- 
onstrate to  him,  as  you  can  that  two  and  two 
make  four,  that  I  am  his  father,  and  that  I  have  a 
right  to  his  obedience.' " l  Thus  did  Pestalozzi  give 
to  arithmetic  an  exaggerated  value  (not  that  the  Pere's 
argument  is  very  convincing),  and  thus  it  assumed  a 
prominence  in  the  curriculum  which  his  followers 
maintained,  and  which  is  only  now,  after  the  lapse  of 
a  century,  being  questioned  by  leading  educators. 

7.  He  emphasized  oral  arithmetic  as  a  mental 
gymnastic,  but  he  unquestionably  carried  the  exer- 
cises too  far.  Knilling,  who  in  his  first  work  wrote 
with  more  force  than  judgment,  was  not  wide  of  the 
mark  when  he  said :  "  The  exercises  with  Pestalozzi's 
Rechentafeln  and  Einheitstabelle  (number  and  units' 
tables)  belong  to  the  most  monstrous,  most  bizarre, 
most  extravagant,  and  most  curious  that  have  ever 
appeared  in  the  realm  of  teaching."2 

1  Payne's  trans,  of  Compayre's  History  of  Pedagogy,  p.  437. 

2  Zur  Reform  des  Rechenunterrichtes,  I,  p.  58.     Those  who  care  to 
know  the  weak  points   of  Pestalozzi,  Grube,  and  other  German   Met/io- 
dikers,  and  to  find  them  discussed  in  vigorous  language,  should  read  this 
work.     The  later  and  more  valuable  works  by  the  same  author  are  also 
worthy  of  study :  Die  naturgemasse  Methode  des  Rechen-Unterrichts  in 
der  deutschen  Volksschule,  I.  Teil,  Miinchen,  18975  II.  Teil,  1899. 


HOW  ARITHMETIC  HAS  BEEN  TAUGHT  85 

8.  He  abandoned  the  mechanism  of  the  old  cipher- 
reckoning,  just  as,  three  centuries  before,  the  cipher- 
reckoners  (algorismists)  had  abandoned  the  abacus,  and 
put  oral  arithmetic  to  the  front.  Number  rather  than 
figures,  was  his  cry.  But  while  instituting  a  healthy 
reaction  against  the  mechanical  rules  of  his  predeces- 
sors, like  most  reformers  he  went  to  the  other  extreme, 
so  much  so  that  the  art  of  ciphering  became  quite 
distinct  from  his  arithmetic.  Against  this  extreme  in 
due  time  another  reaction  set  in  and,  in  America,  drove 
out  the  "mental  arithmetic,"  which  Colburn  had  done 
so  much  to  establish,  replacing  it  by  the  worst  form 
of  mechanism.  In  turn,  against  this  movement  another 
reaction  has  set  in,  and  the  close  of  the  nineteenth 
century  is  seeing  arithmetic  beginning  to  be  placed 
upon  a  much  more  satisfactory  foundation  than  ever 
before. 

Of  Pestalozzi's  contributions  to  arithmetic  but  two 
seriously  influenced  the  world,  perception  as  the  foun- 
dation of  number  teaching,  and  formal  culture  as  the 
aim.  Although  the  creator  of  a  method,  it  found  little 
general  recognition  in  Germany,  and  it  is  known  to-day 
almost  only  by  name.1 

1  Hoose's  Pestalozzian  Arithmetic,  Syracuse,  1882,  made  the  method 
known,  in  its  most  presentable  form,  to  American  teachers.  The  bibliog- 
raphy relating  to  Pestalozzi  is  so  extensive  that  it  is  hardly  worth 
attempting  to  mention  it.  A  brief  resume  of  his  work  is  given  in  Com- 
payre's  History  of  Pedagogy,  and  generally  in  works  of  similar  nature. 
Janicke  gives  the  most  judicial  summary  of  the  conflicting  views  con- 


86     THE  TEACHING  OF  ELEMENTARY   MATHEMATICS 

Tillich  —  Pestalozzi  had  a  host  of  followers  among 
writers  even  though  his  own  method  found  little  favor 
with  teachers.  Among  the  first  of  the  prominent  ones 
was  Tillich,1  who  took  for  his  motto  the  well-known 
but  untranslatable  words,  "Denkend  rechnen  und 
rechnend  denken,"  words  which  might  be  put  into 
English  as :  "  thinkingly  to  mathematize  and  mathe- 
matically to  think."  Acknowledging  the  inspiring  in- 
fluence of  his  master,2  he  nevertheless  saw  the  faults 
of  the  latter's  system  and  boldly  attempted  to  rectify 
them.  His  plan  may  briefly  be  summed  up  as  follows : 

1.  He  paid  much  attention  to  a  systematic  mastery 
of  the  first  decade  of  numbers,  making  this  the  basis 
for  the  advanced  work.     "  My  method  teaches  one  to 
know  all  possible  relations  in  the  first   order  (in  the 
number  space  i-io),  and  by  this  means  to  form  a  stand- 
ard (eine  Norm  bilden)  by  which  all  higher  numbers 
can  be  treated." 

2.  He  did  not  attempt  to  bring  a  child  to  think  of  a 
number,  85  for  instance,  as  so  many  units,  but  rather  as 

cerning  his  theories.  Knilling  is  the  most  interesting  of  his  recent  critics, 
especially  in  his  first  work,  Zur  Reform  des  Rechenunterrichtes,  1884;  "  I 
will,"  he  says,  "  make  it  as  clear  as  day  that  all  the  modern  errors  in  the 
teaching  of  primary  arithmetic  take  themselves  back  to  Pestalozzi,"  —  I, 
p.  2.  On  the  other  hand,  J.  Riiefli  is  Knilling's  most  interesting  critic,  in 
his  work,  Pestalozzi's  Rechenmethodische  Grundsatze  im  Lichte  der  Kri- 
tik,  Bern,  1890. 

1  Allgemeines  Lehrbuch  der  Arithmetik,  oder  Anleitung  zur  Rechen- 
kunst  fur  Jedermann,  1806. 

*  "  Sein  Feuer  hat  mich  entflammt." 


HOW  ARITHMETIC  HAS  BEEN  TAUGHT  87 

so  many  tens  and  so  many  units,  and  similarly  for  larger 
numbers,  —  a  distinct  advance  on  Pestalozzi,  who  failed 
to  bring  out  the  significance  of  the  decimal  system. 

3.  To  bring  out  prominently  this  relation  between 
tens  and  units,  and  between  the  various  units  in  the 
first  decade,  Tillich  devised  what  he  called  a  Reckon- 
ing-chest, a  box  containing  10  one-inch  cubes,  10  paral- 
lelepipeds 2  inches  high  and  an  inch  square  on  the 
base,  10  three  inches  high,  and  so  on  up  to  10  ten 
inches  high.  The  use  to  which  these  rods  were  put 
is  apparent,  and  it  is  also  evident  that  the  ratio  idea 
of  number  was  prominent  in  Tillich's  mind.1 

Of  the  other  followers  of  Pestalozzi,  space  permits 
mention  of  only  two.  Tiirk2  makes  much  of  exercise 
in  thinking,  the  formal  training,3  and  follows  Pesta- 
lozzi in  taking  up  arithmetic  first  without  the  figures 
(in  the  number  space  1-20),  but  he  departs  from  the 
plan  of  his  master  in  not  having  the  child  begin  the 
subject  until  his  tenth  year.  The  formal  culture  idea 
reached  its  height  in  the  works  of  Kawerau;4  his 
extreme  views  provoked  the  reaction. 

1  For  a  modern  treatment  of  the  subject  see  Brautigam's  Methodik  des 
Rechen-Unterrichts,  2.  Aufl.,  Wien,  1895,  P-  4  se(l' 

2  Leitfaden  zur  zweckmassigen  Behandlung  des  Unterrichts  im  Rech- 
nen,  Berlin,  1816. 

8  Uebung  im  Denken,  die  Entwickelung  und  Starkung  des  Denkver- 
mogens. 

*  Leitfaden  far  den  Unterricht  im  Rechnen  nach  Pestalozzischen  Grund- 
satzen,  Bunzlau,  1818. 


88     THE  TEACHING  OF  ELEMENTARY  MATHEMATICS 

Reaction  against  Pestalozzianism  —  It  was  natural 
that  protests  should  arise  against  the  extreme  views  of 
Pestalozzi  and  his  followers.  Like  all  reformers  they 
were  often  intemperate  in  their  demands  and  injudicious 
in  their  plans  for  improvement.  The  reaction  was 
bound  to  come,  and  it  was  led  by  men  of  eminence  in 
educational  affairs,  men  to  whom  we  are  not  a  little 
indebted  for  certain  opinions  now  generally  held. 

For  example,  it  was  Friedrich  Kranckes,  whose  first 
work  appeared  in  1819,  who  suggested  the  four  concen- 
tric circles  which  Grube  afterward  adopted,  exercising 
the  child  in  the  number  space  i-io,  then  in  the  space 
i-ioo,  then  i-iooo,  and  finally  1-10,000.  He,  as  Busse 
had  done  before  him,  employed  number  pictures,  and 
being  one  of  the  best  teachers  in  North  Germany, 
his  influence  greatly  extended  their  use.  He  called  his 
plan  the  Method  of  Discovery  (Erfindungsmethode), 
and  developed  his  rules  from  exercise  and  observation. 
His  problems,  moreover,  were  not  of  the  abstract 
Pestalozzian  type ;  they  touched  the  daily  life  of  the 
child  and  avoided  the  endless  formalism  of  the  Swiss 
master.  Such  common-sense  and  sympathetic  methods 
did  not  fail  to  win  favor  against  Pestalozzi's  frag- 
mentary method. 

Denzel 1  was  another  master  of  the  moderate  school. 
He  laid  down  these  three  aims  in  the  teaching  of  pri- 
mary arithmetic : 

1  Der  Zahlunterricht,  Stuttgart,  1828. 


HOW  ARITHMETIC  HAS  BEEN  TAUGHT  89 

1.  To  exercise  the  thought,  perception,  memory; 

2.  To  lead  the  children  to  the  essence  and  the  simple 
relations  of  number ; 

3.  To  give  the  children  readiness  in  applying  this 
knowledge  to  the  concrete  problems  of  daily  life. 

This  is  a  systematic  and  terse  summary,  and  the  third 
point  is  not  one  which  played  any  part  in  the  Pestaloz- 
zian  scheme.  Denzel,  too,  followed  a  concentric  circle 
plan,  treating  the  four  operations  in  the  circle  i-io, 
then  again  in  the  circle  1-20,  and  so  on. 

Among  the  leaders  who  did  the  most  to  establish 
this  moderate  and  common-sense  school  of  teachers 
must  be  mentioned  Diesterweg1  and  Hentschel,2  men 
whose  opinions  have  done  much  to  mould  the  edu- 
cational thought  of  the  last  half  century. 

Grube  (18 16-1884) 3  —  Grube's  claim  to  rank  as  an 
educator  lies  largely  in  his  power  of  judicious  selection 
from  the  writings  of  others.  He  used  the  "  concentric 
circle"  notion,  but  this  was  half  a  century  old;  he 
made  much  of  objective  work,  but  so  had  every  one 
since  Pestalozzi ;  he  insisted  that  "  every  lesson  in  arith- 
metic must  be  a  lesson  in  language  as  well,"  but  so 
had  Pestalozzi.  He  gave,  however,  one  new  principle, 
—  an  extremely  doubtful  one,  —  that  the  four  funda- 

1  Methodisches  Handbuch   fur  den  Gesammtunterricht  im   Rechnen, 
Elberfeld,  1829. 

2  Lehrbuch  des  Rechenunterrichtes  in  Volksschulen,  1842. 

8  Leitfaden  fur  das  Rechnen,  Berlin,  1842.  Trans,  by  Seeley  (1891), 
and  by  Soldan  (1878). 


9O     THE  TEACHING  OF  ELEMENTARY  MATHEMATICS 

mental  processes  should  be  taught  with  each  number 
before  the  next  number  was  taken  up,1  and  this  is  the 
essence,  the  only  original  feature,  of  the  Grube  method. 

The  book  was  happily  written ;  it  was  brief  —  not  a 
common  virtue ;  it  was  easily  translated,  and  it  thus  be- 
came, some  years  ago,  almost  the  only  German  "method  " 
known  in  America.  Thus  it  has  come  about  that  Grube 
has  been  looked  upon  as  a  name  to  conjure  by,  and 
neither  the  faults  nor  the  virtues  (much  less  the  origi- 
nality) of  the  system  seem  to  have  been  well  considered 
by  most  of  those  who  claim  to  use  it,  —  claim  to,  for 
nobody  actually  does. 

Its  chief  virtue  lies  in  its  thoroughness.  More  than 
a  year  is  given  to  the  number  space  i-io,  and  three 
years  are  recommended  for  the  space  i-ioo. 2  Speak- 
ing of  the  number  space  i-io  he  says :  "  In  the 
thorough  way  in  which  I  wish  arithmetic  taught,  one 
year  is  not  too  long  for  this  important  part  of  the 
work.  In  regard  to  extent  the  pupil  has  not,  appar- 
ently, gained  very  much;  he  knows  only  the  numbers 
from  i  to  10,  —  but  he  knows  them."  There  is,  how- 
ever, such  a  thing  as  being  too  thorough;  to  know 
all  that  there  is  about  a  number  before  advancing  to 
the  next  one  is  as  unnecessary  as  it  is  illogical,  as 

1  Allseitige  Zahlenbehandlung. 

2  See  the  6th  (last)  edition  of  the  Leitfaden,  1881,  p.  25,  n. :  "  Al- 
ways from  the  educational  standpoint  one  must  extend  the  .first  course 
(*'.*.,  i-ioo)  over  three  years  for  the  majority  of  pupils." 


HOW  ARITHMETIC  HAS  BEEN  TAUGHT  91 

impossible  as  it  is  uninteresting.  Instead  of  requir- 
ing more  time  for  the  group  i-io  when  he  published 
his  sixth  edition  (1881)  than  he  did  when  he  pub- 
lished the  first  (1842),  Grube  might  well  have  re- 
quired less.  Home  training  and  the  training  of  the 
street  are  such  that  children  know  more  about  num- 
bers now  than  they  did  in  the  first  hah0  of  the  cen- 
tury. The  interesting  studies  of  Hartmann,  Tanck, 
and  Stanley  Hall  have  shown  that  most  children  have 
a  very  fair  knowledge  of  numbers  to  five  before  en- 
tering school.  On  the  other  hand,  of  course  the  ability 
to  count  must  not  be  interpreted  to  mean  that  the  child 
has  necessarily  any  clear  notion  of  number.  Children 
often  count  to  100,  as  their  elders  often  read  poetry, 
with  little  attention  to  or  appreciation  of  the  meaning. 
The  chief  defects  of  the  system  are  these: 

1.  It  carries   objective   illustration   to   an   extreme, 
studying    numbers    by   the    aid    of    objects    for   three 
years,  until  100  is  reached.1 

2.  It  attempts  to   master  each   number  before  tak- 
ing up  the  next,  as  if  it  were  a  matter  of  importance 
to   know  the  factors  of    51    before   the  child   knows 
anything   of   75,   or    as    if    it   were   possible   to   keep 
children   studying  4   when   the   majority  know   some- 
thing of  8  before  they  enter  school. 

3.  It  attempts  to  treat  the  four  processes  simulta- 

1  On  the   proper  transition  from  the   concrete  to  the  abstract,  see 
Payne's  trans,  of  Compayre's  Lectures  on  Pedagogy,  p.  384. 


92     THE  TEACHING  OF  ELEMENTARY  MATHEMATICS 

neously,  as  if  they  were  of  equal  importance  or  of 
equal  difficulty,  which  they  are  not. 

While  all  must  recognize  that  Grube  gives  many  val- 
uable suggestions  to  teachers,  the  system  as  set  forth 
in  the  last  edition  of  the  Leitfaden  has  almost  no  sup- 
porters. "  While  stimulating  to  children  if  not  carried 
to  excess,  it  easily  degenerates  into  mere  mechanism,  as 
every  one  will  agree  who  has  carefully  looked  into  it." 1 

Of  the  later  "methods,"  but  two  or  three  can  be 
mentioned.  Kaselitz2  has  criticised  his  predecessors 
by  saying  that  they  teach  a  great  deal  about  number, 
/7/but  do  not  teach  the  child  how  to  operate  with  num- 
ber. He  therefore  develops,  and  with  much  skill,  the 
idea  of  making  the  number  the  operator. 

Knilling3   and   Tanck4  are   leaders   in   the   modern 

1  Dittes,  Methodik  der  Volksschule,  205.     "  Ein  Instrument  mit  dem 
nur  Meister  umgehen  konnen."  —  Earth  olomai.     "  Unmoglich,  langweilig, 
zeitraubend,  und  ganz  unniitz.  .  .  .     Die  Behandlung  jeder  Einzelzahl  1st 
unmoglich  und  auch  vollig  unniitz."  —  Kallas,  Die  Methodik  des  elemen- 
taren  Rechenunterrichts,  Mitau,  1889,  p.  20,  22.     A  good  summary  of  the 
system  is  given  in  Unger,  p.  188-195.      An  earnest  protest  against  the 
whole  system  is  set  forth  in  Zwei  Abhandlungen  uber  den  Rechenunter- 
richt,  by  Christian  Harms,  Oldenburg,  1889.     The  method  is  known  to 
American  teachers  through  translations  of  the  earlier  editions,  made  by 
Soldan  and  by  Seeley. 

2  Wegweiser  fur  den  Rechenunterricht  in  deutschen  Schulen,  Berlin, 
1878,  and  other  works. 

8  Works  already  cited.  For  brief  review  see  Hoffmann's  Zeitschrift, 
XXVIII.  Jahrg.,  p.  514. 

4  Rechnen  auf  der  Unterstufe,  1884  ;  Der  Zahlenkreis  von  I  bis  20, 
Meldorf,  1887  ;  Betrachtungen  uber  das  Zahlen,  Meldorf,  1890. 


HOW  ARITHMETIC  HAS  BEEN  TAUGHT  93 

pre-Pestalozzian  movement.  They  assert  that  from 
Pestalozzi  to  the  present  time  teachers  have  been 
assuming  that  number  is  the  subject  of  sense-percep- 
tion, which  it  is  not.  "Number  is  not  (psychologi- 
cally) got  from  things,  it  is  put  into  them." J  They 
proceed  to  base  their  system  upon  the  counting  of 
things,  a  process  in  which  three  ideas  are  prominent, 
(i)  the  counted  mass,  (2)  the  how  many,  (3)  the 
sense  in  which  the  things  are  considered.  Knilling2 
classifies  the  numbers  of  arithmetic  as  (i)  numbers 
of  natural  units  —  as  of  things,  men,  trees,  etc. ;  (2) 
numbers  of  measured  units  —  as  of  metres,  grammes, 
etc. ;  (3)  numbers  of  mathematical  units.  The  mathe- 
matical unit  is  without  quality  (color,  form,  etc.);  it 
is  without  extent;  it  is  indivisible,  a  notion  going 
back  to  Aristotle ;  it  occupies  no  space ;  it  is  not 
imageable.  Such  a  unit  does  not  exist  in  the  external 
world;  it  exists  only  in  the  mind. 

The  child  likes  to  count;  the  rhythm  of  counting 
is  pleasing.3  "The  fact  that  at  least  nearly  all  chil- 
dren, no  matter  how  taught,  first  learn  to  count  in- 
dependently of  objects,  in  which  the  series  idea  gets 
ahead,  —  that  they  recognize  three  or  four  objects  at 
first  as  individuals,  calling  the  fourth  one  four  even 
when  set  aside  by  itself,  —  that  counting  proceeds  in- 

1  McLellan  and  Dewey,  The  Psychology  of  Number,  p.  61. 

*  Die  naturgemasse  Methodik,  I,  p.  55. 

•  Phillips,  D.  E.,  Pedagogical  Seminary,  V,  p.  233. 


94     THE  TEACHING  OF  ELEMENTARY  MATHEMATICS 

dependently  of  the  order  of  number  names,  and  often 
consists  in  a  repetition  of  a  few  names  as  a  means 
of  following  the  series,  —  that  children  desire  and 
learn  these  names,  —  such,  taken  with  the  earlier 
steps  presented,  furnish  unmistakable  evidence  that 
the  series  idea  has  become  an  abstract  conception. 
.  .  .  The  naming  of  the  series  generally  goes  in 
advance  of  its  application  to  things,  and  the  ten- 
dency of  modern  pedagogy  has  been  to  reverse  this. 
.  .  .  Counting  is  fundamental,  and  counting  that  is 
spontaneous,  free  from  sensible  observation  and  from 
the  strain  of  reason.  ...  In  the  application  of  the 
series  to  things  is  where  the  child  first  encounters 
much  difficulty,  and  this  is  much  increased  because 
the  teacher,  not  apprehending  the  full  importance  of 
this  step,  tries  to  hurry  the  child  over  this  point 
entirely  too  rapidly.  It  is  here  that  we  meet 
with  so  many  systems  and  devices  for  teaching 
numbers." 1 

Upon  this  natural  desire  to  count,  Knilling  and 
Tanck  base  their  method,  a  systematic  arrangement 
of  counting  forward  and  backward  by  ones,  twos, 
etc.,  within  the  first  hundred,  leading  easily  to  rapid 
work  in  addition,  subtraction,  multiplication,  and  even 
division.  Mental  pictures  of  numbers  are  of  no  value 
in  actual  work ;  all  calculation  is  figure  work ;  the 
head  is  never  more  empty  of  mental  pictures  than 

1  Phillips,  D.  E.,  Pedagogical  Seminary,  V,  p.  221. 


HOW  ARITHMETIC  HAS  BEEN  TAUGHT  95 

when  we  calculate;  calculation  is  not  a  matter  of 
perception,  it  is  a  mechanical  affair  pure  and  simple. 

But  given  these  exercises  in  running  up  and  down 
the  numerical  scale,  one  is  no  nearer  being  an  arith- 
metician than  is  one  who  can  finger  the  scales  on  the 
piano  to  being  a  musician.  Each  furnishes  the  best 
basis  for  subsequent  work  and  skill.1 

One  of  the  most  temperate  of  writers  upon  this  phase 
of  number  work2  thus  summarizes  the  discussion  : 

1.  Since  through  language  number  space  was  first 
created,   and   since   here   lies  the   source   of   all  com- 
putation, therefore  the  teacher  must  impress  upon  the 
child  the  sequence   of  number  words  as  a  true,  ser- 
viceable and  lasting  sound  series  (Lautreihe). 

2.  Since  with  this  series  must  in  due  time  be  asso- 
ciated things,  perception  enters. 

3.  Since  the  number  words  establish  only  the  chron- 
ological difference  in  the  appearance  of  the  individual 
units,  suitable  exercises  should  be  given  to  make  the 
pupil  certain  as  to  his  order  of  the  units. 

This  relation  of  number  to  time  (sequence)  is  not 
new,  and  the  subject  has  been  a  ground  for  debate 
since  Kant  first  made  it  prominent.  Sir  William 
Hamilton  takes  one  side  and  talks  about  "the  science 
of  pure  time."  Herbart8  on  the  other  hand  main- 

1  "  Diese  Uebungen  sind  so  wenig  das  Rechnen  selbst,  als  Uebungen  in 
den  Scalen  und  in  den  Intervallen  die  Musik  sind."     Fitzga,  p.  23. 

2  Fahrmann,  K.  Emil,  Das  rhythmische  Zahlen,  Plauen  i.  V,  1896,  p.  24. 
*  Psychologic  als  Wissenschaft,  II,  p.  162. 


OF  THE 

dNiVERSITY 


96     THE  TEACHING  OF  ELEMENTARY  MATHEMATICS 

tains  that  number  is  no  more  related  to  time  than  to 
a  hundred  other  concepts.  Lange  relates  number  to 
space  rather  than  to  time,  saying,  "  The  oldest  ex- 
pressions for  number  words  relate,  so  far  as  we  know 
their  meaning,  to  objects  in  space.  .  .  .  The  alge- 
braic axioms,  like  the  geometric,  refer  to  space-per- 
ceptions."1 "Every  number  concept  is  originally  the 
mental  picture  of  a  group  of  objects,  be  they  fingers 
or  the  buttons  of  an  abacus."2  On  the  other  hand, 
Tillich,  whose  method  does  not  wholly  agree  with  his 
sentiment,  thus  sets  forth  his  views  upon  this  point: 
"  The  empirical  of  arithmetic  is  to  be  sought  in  Time 
alone.  It  is  therefore  only  the  number  arrangement 
which  is  capable  of  representation  to  the  senses,  and 
only  the  sequence  which  must  be  fixed  in  the  first 
exercises,  for  from  this  everything  else  develops.  .  .  . 
Number  has  nothing  spatial  about  it,  it  exists  only  in 
Time,  and  not  as  anything  absolute  there,  but  only 
as  something  relative.  The  sequence  is  the  great 
thing,  not  the  magnitude."  3 

This  return  to  the  pre-Pestalozzian  idea  of  begin- 
ning with  exercises  in  counting — but  in  a  much  more 
systematic  way  than  any  of  Pestalozzi's  predecessors 
followed  —  is  the  latest  phase  of  instruction  in  arith- 
metic which  has  commanded  very  general  attention. 

1  Logische  Studien,  p.  140. 

2  Geschichte  des  Materialismus,  II,  p.  26. 
8  Lehrbuch  der  Arithmetik,  p.  331,  333. 


HOW  ARITHMETIC  HAS  BEEN  TAUGHT  97 

The  idea  has  been  presented  in  America  by  Phillips.1 
But  in  working  out  the  method  in  detail,  the  German 
writers  have  gone  to  an  extreme,  assigning  "alto- 
gether too  much  value  to  counting  —  and  to  counting 
in  a  narrow  sense,  mere  memory  work  with  the  num- 
ber series  without  reference  to  real  things.  ...  It 
is  a  great  overrating  of  the  value  of  counting.  .  .  . 
Counting  should  be  the  servant  of  number  work,  not 
number  work  the  servant  of  counting."2 

1  Some  Remarks  on  Number  and  its  Applications,  Clark  University 
Monograph,  1898;  Number  and  its  Applications  Psychologically  consid- 
ered,  Pedagogical  Seminary,  October,  1897. 

a  Grass,  J.,  Die  Veranschaulichung  beim  grundlegenden  Rechnen, 
Munchen,  1896,  p.  10. 


CHAPTER  V 
THE  PRESENT  TEACHING  OF  ARITHMETIC 

Objects  aimed  at  —  In  Chapter  IV  the  growth  of 
the  teaching  of  primary  arithmetic  was  briefly  traced. 
The  teaching  of  the  more  advanced  portions  was  not 
considered.  In  the  present  chapter  a  few  of  the  recent 
tendencies  in  both  primary  and  secondary  arithmetic 
will  be  briefly  mentioned,  and  chiefly  with  a  view  to 
ascertaining  what  are  a  few  of  the  points  of  con- 
troversy. 

In  the  first  place,  it  is  not  at  all  settled  as  to  what 
we  are  seeking  in  teaching  arithmetic  to  a  child. 
Herbart  and  his  followers  would  have  us  bring  out 
the  ethical  value.  Others  equally  prominent  and  more 
numerous  assert  that  it  has  no  such  value.  "We  en- 
tirely overrate  arithmetic  if  we  ascribe  to  it  any 
soul-forming  ethical  power.  .  .  .  The  mental  activity 
(Denkthatigkeit)  induced  by  arithmetic  is  unproduc- 
tive and  heartless  (gemiitlos)." l  Grube  and  many 
others  would  make  it  adapt  itself  to  language  work, 
Pestalozzi  made  much  of  the  logical  training  which 
it  gave,  and  several  writers  have  amused  themselves 

1  Korner,  Geschichte  der  Padagogik,  1857. 
98 


THE   PRESENT  TEACHING  OF  ARITHMETIC  99 

by  giving  quite  extended  lists  of  divers  virtues  cul- 
tivated by  the  simple  science  of  numbers. 

But  it  sometimes  seems  as  if  these  discussions  have 
been  more  harmful  than  beneficial.  When  we  hear 
some  second  year  class  dawdling  along  through  a 
little  simple  number  work,  which  no  doubt  has  been 
elegantly  developed,  and  out  of  which  ethical  and 
logical  and  general  culture  values  have  no  doubt  been 
duly  extracted,  we  are  forced  to  wonder  whether  in 
a  maze  of  secondary  purposes  there  is  not  lost  the 
primary  purpose  —  that  of  leading  the  child  to  "figure" 
quickly  and  accurately  in  the  common  problems  of 
his  experience. 

The  number  concept  —  The  fundamental  principle  in 
the  method  of  teaching  primary  arithmetic  has  its 
root  in  the  essence  of  number.1  No  one  now 
affirms  that  number  is  an  object  of  sense-perception,2 
although  upon  this  inherited  notion  are  based  not  a 
few  of  our  present  ideas  as  to  method.  "The  notion 
of  number  is  not  the  result  of  immediate  sense-per- 
ception, but  the  product  of  reflection,  of  an  activity  of 
our  minds.  We  cannot  see  nine.  We  can  see  nine 
horses,  nine  feet,  nine  dollars,  etc.,  that  is  we  see 
the  horses,  the  feet,  the  dollars,  if  they  are  presented 
to  us;  that  there  are  exactly  nine,  however,  we  cannot 

1  Beetz,  K.  O.,  Das  Wesen  der  Zahl  als  Einheitsprinzip  im  Rechen- 
unterricht     Neue  Bahnen,  VI.  Jahrg.,  201. 

2  McLellan  and  Dewey,  p.  61. 


100    THE  TEACHING  OF  ELEMENTARY  MATHEMATICS 

see.  If  we  wish  to  know  this  we  are  forced  to  count 
the  things ;  and  since  we  usually  do  this  with  the  help 
of  our  eyes,  the  idea  has  got  abroad  that  we  see 
number."  1 

In  line  with  this  idea  we  would  be  justified  in  say- 
ing that  one  is  not,  primarily,  a  number,  and  it  is 
historically  interesting  to  know  that  only  recently  has 
it  been  so  considered.  The  classical  definition  of 
number  is  "a  collection  of  units," 2  a  definition  scien- 
tifically worthless. 

But  while  we  put  number  into  objects,  on  the  other 
hand  we  derive  our  idea  of  number  only  from  the 
presence  of  the  world  external  to  the  mind.  We  see  a 
group  of  people,  and  we  begin  by  making  an  abstrac- 
tion ("  people  "),  and  we  say,  "  Here  are  ten  people  "  — 
thus  calling  them  all  by  the  one  abstract  name,  even 
though  the  individuals  be  very  different.  "A  careful 
observation  shows  us,  however,  that  there  are  no 
objects  exactly  alike;  but  by  a  mental  operation  of 
which  we  are  quite  unconscious,  although  it  holds 
within  itself  the  entire  secret  of  mathematical  ab- 
straction, we  take  in  objects  which  seem  to  be  alike, 

1  Fitzga,  E.,  Die  natiirliche   Methode  des  Rechen-Unterrichtes  in  der 
Volks-  und  Biirgerschule,  I.  Theil,  Wien,  1898.     This  is  one  of  the  most 
common-sense  books  on  method  that  has  appeared  in  a  long  time. 

2  This  is  found  in  most  of  the  older  arithmetics.     For  example,  Gemma 
Frisius,  in  his  famous  text-book,  says,  "  Numerum  autores  vocant  multitu- 
dinem  ex  unitatibus  conflatum.     Itaque  unitas  ipsa  numerus  non  erit." 
Arithmeticae  Practicae  Methodus  Facilis,  Witebergae,  M.D.  LI,  pars  prima. 


THE  PRESENT  TEACHING  OF  ARITHMETIC          IOI 

rejecting  for  the  time  being  their  differences.  Here 
is  to  be  found  the  source  of  calculation."1  So  the 
idea  of  number  is  generated  in  the  mind  by  the  sense- 
perception  of  a  group  of  things  supposed  to  be  alike.2 
Hence  while  we  do  not  have  a  sense-perception  of 
number,  on  the  other  hand  few  now  attempt  to  teach 
number  without  the  help  of  objects  for  the  formation 
of  groups.  What  these  objects  shall  be  is  more  of  a 
dispute  to-day  than  ever  before.  In  Germany  the  use 
of  numeral  frames  has  been  carried  to  an  extent  not 
known  in  America,  and  several  forms  of  apparatus 
have  been  devised.  But  however  valuable  these  aids 
may  be  in  the  first  grade,  it  is  doubtful  if  there  is 
any  excuse  for  their  extensive  use  thereafter.3  In 
America  the  tendency  has  been  along  the  Pestaloz- 
zian  line,  of  taking  any  material  that  is  at  hand, 
although  objection  has  been  made  to  the  most 
natural  means  of  all,  the  fingers.4  Frequently,  how- 

1  Laisant,  La  Mathe*matique,  p.  15,  1 8,  19,  31. 

2  "  Jede  Zahl  ist  der  Inbegriff  einer  gewissen  Menge  von  Einheiten. 
Einheiten   im   Sinne   des   ersten   Rechnens  sind  wirkliche    Dinge.  .  .  . 
Ein   grundlegender   Rechenunterricht   ohne  Veranschaulichung   ist  ... 
undenkbar."      Grass,  J.,   Die   Veranschaulichung   beim    grundlegenden 
Rechnen,  Munchen,  1896,  p.  5,  6. 

8  One  of  the  best  brief  historical  discussions  of  numeral  frames  is  given 
in  Grass,  op.  cit.,  61  seq.  The  matter  is  discussed  in  Payne's  transl.  of 
Compayre's  Lectures  on  Pedagogy,  p.  384-385,  the  note  on  p.  385  being 
misleading,  however. 

4  Die  Finger  sind  das  natiirlichste  und  nachste  Versinnlichungs- 
mittel.  Fitzga,  I,  p.  18. 


102    THE  TEACHING  OF  ELEMENTARY  MATHEMATICS 

ever,  teachers  have  fallen  into  the  error  of  forgetting 
Busse's  valuable  suggestion,  that  the  objects  should 
not  be  such  as  to  take  the  child's  attention  from  the 
central  thought.  At  the  same  time,  they  should  be 
such  as  relate  to  his  daily  life  and  such  as  have 
some  interest  for  him.1 

There  has  also  been  a  tendency  in  America  to 
follow  Grube  to  the  extreme  of  using  objects  long 
after  there  is  any  need  for  them.  Some  have 
devoted  much  energy  to  bringing  children  to  recog- 
nize at  a  glance  the  number  in  a  group,  say  nine, 
and  this  has  connected  itself  with  the  best  form  of 
grouping  to  establish  number  relations  and  to  enable 
the  eye  to  grasp  the  group  readily.  A  consideration 
of  the  forms 


•  •  I  •  • 


I  J       •  •  •  • 


.-... 


shows  how  much  more  readily  the  eye  grasps  some 
forms  than  others.  But  after  all,  this  is  fundamen- 
tally the  recognition  of  a  familiar  form,  which  we 
have  learned  has  a  certain  number  of  spots,  rather 
than  the  recognition  of  a  number.  In  a  game  of 

1  Was  dutch  das  Leben  in  Schule  und  Haus  und  ausser  dem  Hause 
in  den  Erfahrungskreis  des  Kindes  gekommen  ist,  auch  das  kann  fur  das 
Rechnen  verwertet  werden.  Alle  Teile  des  Gedankenkreises  sollen  rech- 
nerisch  durchleutet  werden,  in  denen  ihrer  Natur  nach  Zahlen  eine  Rolle 
spielen.  Rein,  Pickel  and  Scheller,  Theorie  und  Praxis  des  Volksschul- 
unterrichts,  I,  p.  361. 


THE  PRESENT  TEACHING  OF  ARITHMETIC          103 

cards  we  recognize  the  form  of  the  nine  as  we  do 
the  form  of  the  knave ;  we  do  not  stop  to  count  the 
spots,  nor  could  we  tell  the  number  on  a  different 
arrangement  unless  we  counted.1 

The  uselessness  of  carrying  this  objective  work  too 
far  is  apparent  when  we  consider  that  we  never  get 
our  ideas  of  numbers  of  any  size  from  thinking  of 
groups;  we  get  them  from  thinking  of  the  relative 
places  which  they  occupy  in  the  number  series,  or  ,> 
the  time  which  it  takes  to  reach  that  place  in  run- 
ning up  that  series,  or  the  length  of  the  line  which 
would  represent  that  number  in  comparison  with  unity.3 

Recently,  sustained  by  high  psychological  authority, 
the  effort  has  been  made  to  make  prominent  the  ratio 
idea  from  the  very  outset.  That  ratio  is  number  is 
evident;  that  the  converse  is  true,  has  the  authority 
of  Newton's  well-known  definition ;  that  a  child  should 
first  consider  number  in  this  way  has  its  advocates. 
"  The  fundamental  thing,"  says  one  of  these  "  (in 
teaching  arithmetic),  is  to  induce  judgments  of  rela- 
tive magnitudes."3  But  such  a  scheme  substitutes  a 

1  If  one  cares  to  enter  this  field  with  any  thoroughness,  historically  and 
psychologically,  he  should  read  Grass,  op.  cit.,  p.  14  seq.,  one  of  the  best 
discussions  available. 

2  Um  uns  grossere  Zahlen  ohne  Wiederholung  des  Zahlens  etwas  deut- 
licher  zu  vergegenwartigen,  greifen  wir  daher  zu  dem  Auskunftsmittel  von 
Substitutionen.     Das  gebrauchlichste   ist,  fdr  Zahlvorstellungen  Zeitvor- 
stellungen  zu  substitutieren.     Fitzga,  I,  p.  16. 

*  Speer,  W.  W.,  The  New  Arithmetic,  Boston,  1896. 


104    THE  TEACHING  OF   ELEMENTARY  MATHEMATICS 

complex  for  a  simple  number  idea,  it  is  contrary  to 
the  historical  sequence  (whatever  that  may  be  worth), 
and  it  makes  use  of  a  notion  of  number  entirely  dif- 
ferent from  that  of  which  the  child  will  be  conscious 
in  his  daily  life.  It  founds  the  idea  of  number  upon 
measurement,  but  in  so  doing  it  uses  the  word  measure 
in  its  narrowest  sense.  It  makes  use,  also,  of  sets  of 
objects  (in  the  systems  thus  far  suggested)  by  which 
is  accomplished  no  more  than  Tillich  accomplished 
with  his  blocks,  while  their  character  is  such  as  to 
take  the  attention  from  the  central  thought  of  number. 
Fundamentally,  as  Laisant  has  pointed  out,  and 
Comte  before  him,  the  two  notions  of  counting  and 
measuring  are  the  same.1  The  estimation  of  a  mag- 
nitude directly  by  comparison  is,  however,  extremely 
rare ;  "  it  is  the  indirect  measure  of  magnitudes  which 
characterizes  mathematics."  As  to  the  necessity  for 
the  ratio  idea  at  some  time  in  the  pupil's  course, 
there  can  be  no  question ;  the  argument  lies  only  as 
to  where  the  idea  should  be  brought  in.2  The  most 
temperate  and  philosophical  discussion  of  the  subject 
is  that  given  by  McLellan  and  Dewey  in  their  "  Psy- 
chology of  Number"  (1895),  a  work  which  should  be 
read  and  owned  by  every  teacher  in  the  elementary 
grades.  It  makes  number  depend  upon  measurement, 
but  it  uses  this  word  in  the  broader  sense  indicated 

1  Laisant,  La  Mathematique,  p.  17. 

2  A  brief  but  very  good  discussion  is  given  in  Beetz,  op.  cit,  p.  299. 


THE  PRESENT  TEACHING  OF  ARITHMETIC          105 

by  Comte,  including  counting  as  a  special  form.  In 
counting,  however,  it  wages  war  against  the  "  fixed 
unit"  system  which  the  authors  brand  with  Grube's 
name,  although  Grube  is  by  no  means  the  father  of 
it.  It  actually  (as  all  do  theoretically)  substitutes  the 
method  of  things  for  the  method  of  symbols,  the 
Pestalozzian  idea  of  numbers  instead  of  figures,  and 
it  leads  a  general  attack  against  the  inherited  weak- 
nesses of  the  traditional  primary  arithmetic.  The 
work  seems  not  to  seek  to  place  upon  the  child  the 
burden  of  the  ratio  idea  at  the  outset,  but  rather  to 
lead  him  to  a  common-sense  notion  of  number  with- 
out fixed  unit,  of  counting  in  the  best  form  of  the 
Knilling-Tanck  school,  of  applying  the  knowledge  of 
number  to  things  instead  of  to  relations  of  volumes 
and  lengths.  To  count  things ;  not  to  say  3  +  5  =  ?, 
but  3  cts.  +  5  cts.  =  how  many  cents  ?,  or  3  five-cent 
pieces  4-  5  five-cent  pieces  are  how  many  five-cent 
pieces  ?  —  this  is  to  use  number  as  the  world  first 
used  it,  to  use  number  with  a  varying  unit,  to  get  an 
introduction  to  ratio  at  the  best.1  Laisant  sums  up 
the  matter  of  the  proper  place  for  the  ratio  idea 
when  he  says :  "  It  is  proper  to  ask  if  the  idea  of 
ratio,  usually  assigned  place  rather  late  in  the  study 
of  arithmetic,  does  not  deserve  to  be  considered  early 
in  the  course  as  a  consequence  of  the  notion  of  number"* 

1  Fitzga,  p.  28  ;  McLellan  and  Dewey,  p.  78,  147,  149,  etc. 
*  La  Mathematique,  p.  30. 


106    THE  TEACHING  OF  ELEMENTARY  MATHEMATICS 

When  in  elementary  work  we  are  led  to  feel  that 
a  child  must  not  only  think  of  a  group  of  things  or 
a  ratio  when  he  is  learning  about  the  numbers  from 
i  to  10,  but  that  he  must  continue  to  think  of  groups 
and  ratios,  and  to  refer  to  objects,  as  he  progresses, 
we  impose  upon  him  what  no  mathematician  takes 
upon  himself.  The  child  must  get  his  first  notion  of 
numbers  from  counting  things,  as  the  world  did ;  these 
things  may  in  themselves  be  groups;  in  counting  he 
really  measures  the  group  by  the  unit  with  which  he 
is  working;  he  gets  a  ratio,  if  we  please  to  call  it  so, 
although  the  concept  is  not  simple  enough  to  be  thrust 
upon  him.  But  once  the  idea  of  number  is  there,  it 
is  then  largely  a  matter  of  the  number  series ;  we  have 
an  idea  of  forty-seven  as  lying  between  forty-six  and 
forty-eight,  a  little  below  fifty,  and  as  being  a  number 
about  half  way  (distance)  to  a  hundred,  and  we  have  a 
vague  idea  that  it  would  not  take  long  to  count  it, 
about  half  as  long  (time)  as  to  count  a  hundred.  Thus 
we  place  it  in  a  series,  on  a  line,  or  in  the  flow  of 
time,  and  thus  we  get  an  idea  of  its  magnitude;  but 
few  people  visualize  it  as  a  group  of  objects,  and  why 
should  a  child  be  asked  to  do  so? 

Advocates  of  the  idea  that  number  means  merely 
the  how-many  of  a  group,  or  the  ratio  of  lengths 
merely,  are  disappearing  as  such  scientific  writers  as 
Grassmann,  Hankel,  G.  Cantor,  and  Weierstrass  are 
coming  to  be  known.  The  doctrine  of  "one-to-one 


THE  PRESENT  TEACHING  OF  ARITHMETIC          107 

correspondence"  is  being  understood  by  elementary 
teachers,  and  it  is  not  without  suggestiveness  in  simple 
work  in  arithmetic.  To  the  number  of  a  group  cor- 
responds one  name  and  one  symbol,  as 

*•*  five  5 

If  we  establish  the  laws  of  these  numbers,  as  that 

•  •  •  •  •    • 

•      and      •      equal      •      and      • 

•  •  •  •  •    • 

and  give  to  a  certain  operation  one  name  and  one 
symbol  (as  "addition,"  -f  ),  then  we  may  work  with 
symbols  according  to  these  laws,  and  we  need  have 
no  thought  of  the  names  or  the  numbers,  but  can 
translate  back  into  numbers  at  any  time  we  choose. 
Indeed,  our  symbols  may  force  us  to  establish  new 
kinds  of  numbers,  as  when  we  run  up  against  the 
symbols  4  —  6,  or  V4>  °r  trv  to  divide  the  circumfer- 
ence of  a  circle  by  the  diameter.  This  notion  of  "one- 
to-one  correspondence,"  while  not  consciously  one  of 
elementary  arithmetic,  exists  there  just  as  really  as  it 
exists  in  later  work.  It  does  not  take  long  for  the 
child  to  "substitute  for  the  reality  of  things  the 
creatures  of  reason,  born  of  his  own  mind."  In  solv- 
ing a  problem,  be  it  one  in  the  calculus,  in  algebra, 
or  in  the  second  year  of  arithmetic,  we  begin  by  sub- 
stituting for  the  actual  things  certain  abstractions 
represented  by  symbols;  we  think  in  terms  of  these 


108    THE  TEACHING  OF  ELEMENTARY  MATHEMATICS 

abstractions,  aided  by  symbols,  and  finally  from  our 
result  we  pass  back  to  the  concrete  and  say  that  we 
have  solved  the  problem.  It  is  all  a  matter  of  "  one- 
to-one  correspondence,"  it  being  easier  for  us  to  work 
with  the  abstract  numbers  and  their  corresponding 
figures  than  to  work  with  the  actual  objects.  Funda- 
mentally the  process  is  something  like  this : 

1.  By  abstraction  we  pass  to  numbers. 

2.  Thence  we  pass   to  symbols,  and  we    make    an 
equation,    either  openly,  as   in    algebra,  or   concealed, 
as  in   many   forms   of   arithmetic.     This   equation  we 
solve,  the  result  being  a  symbol. 

3.  We  find  the  number  corresponding  to  this  sym- 
bol, and  say  that  the  problem  is  solved. l 

All  this  does  not  mean  that  primary  number  is  to 
be  merely  a  matter  of  symbols.  It  means  that  in 
mathematics  we  find  it  more  convenient  to  work 
purely  with  symbols,  translating  back  to  the  corre- 
sponding concrete  form  as  may  be  desired.  And  so 
those  teachers  who  fear  lest  the  child  shall  drift  into 
thinking  in  symbols  instead  of  in  number,  are  really 
fearing  that  the  child  shall  drift  into  mathematics. 
In  a  rough  way  we  may  summarize  the  conclusions 
of  the  writers  to  whom  reference  has  chiefly  been 
made,  as  follows : 

i.  Let  the  child  learn  to  count  things,  thus  getting 
the  notion  of  number.  These  things  are,  for  the  pur- 

1  Laisant,  La  Mathematique,  p.  20,  21. 


THE  PRESENT  TEACHING  OF  ARITHMETIC         109 

pose  of  counting,   considered   alike,  and   they  may  be 
single  objects  or  groups. 

2.  Let    him    acquire   the    number   series,  exercising 
with  it  beyond  the  circle   of   actually  counted   things. 

3.  In  the  learning   of   symbols  it  does  not  seem  to 
be  a  matter  of  moment  as  to  whether  these  are  given 
with  the  first   presentation   of  number   or   not.     They 
must,  however,  be  acquired  soon. 

4.  Unconsciously    and    gradually     the     child     will 
acquire  the   idea  (never   expressed   to   him  in   words) 
of   the    one-to-one    correspondence    of    number,    name, 
symbol,   and   thereafter  the   pure   concept   of   number 
will  play  a  small  part  in  his  arithmetical  calculations. 

5.  The  ratio  idea  of   number  should   be  introduced 
early,  and  applied  in  the  work  with  fractions. 

The  great  question  of  method — M.  Laisant  has  tersely 
expressed  what  is  probably  in  the  minds  of  most  suc- 
cessful teachers  qf  elementary  mathematics,  in  the 
following  words :  "  There  are  not,  I  believe,  many 
methods  of  teaching,  if  by  teaching  we  are  to  under- 
stand the  ensemble  of  efforts  by  which  we  seek  to 
furnish  with  accurate  knowledge  a  human  mind  which 
has  not  yet  reached  its  full  degree  of  development. 
.  .  .  The  problem  is  always  the  same:  —  to  interest 
the  pupil,  to  induce  research,  to  continually  give  him 
the  notion,  the  illusion  if  you  please,  that  he  is  dis- 
covering for  himself  that  which  is  being  taught  him."  * 

1  La  Mathematique,  p.  1 88,  189. 


1 10    THE  TEACHING  OF   ELEMENTARY   MATHEMATICS 

As  for  the  rest,  it  is  largely  a  matter  of  psycho- 
logical presentation  and  detailed  device.  Shall  we 
extract  square  root  by  the  diagram  or  by  the  formula? 
—  The  question  is  of  relatively  little  importance  in 
comparison  with  the  great  questions  of  method  and 
of  psychological  presentation.  So  with  most  of  the 
questions  to  be  discussed  in  this  chapter;  they  are 
matters  of  detail  which  one  teacher  may  work  out 
one  way,  and  another  a  different  way,  and  the  differ- 
ence in  result  may  be  so  slight  that  the  world  has 
not  been  able,  after  centuries  of  experiment,  to  decide 
which  is  better.  These  matters  vary  with  classes, 
with  the  advancement  of  pupils,  and  with  the  temper- 
ament of  the  teacher.  To  give  simplicity  of  form  with' 
depth  of  thought  is  one  of  the  qualities  of  the  diffi- 
cult art  of  teaching,  and  it  depends  upon  the  individ- 
ual to  attain  to  this  simplicity.1 

The  advance  in  the  modern  teaching  of  arithmetic 
is  due  much  more  to  the  recognition  of  the  definite 
aim  than  to  the  discovery  of  improved  methods.  On 
the  other  hand,  the  influence  of  such  writers  as  De 
Gar  mo  and  the  McMurrys  in  America,  opening  up 

1  "  Les  moyens  materials,  les  precedes  pedagogiques  a  mettre  en 
oeuvre  pour  obtenir  le  resultat  desire  sont  eminemment  variables,  suivant 
la  nature  des  classes,  1  'avancement  des  eleves,  et  aussi  d'apres  la 
maniere  de  voir  et  le  temperament  du  professeur.  .  .  .  Cette  concilia- 
tion de  la  simplicite  dans  la  forme  avec  la  profondeur  des  ide"es  con- 
stitue  1'une  des  qualites  de  1'art  difficile  de  1 'enseignement."  Laisant, 
p.  192,  194. 


f  V 

j  OF  TH 

1  UN1VE 


THE  PRESENT  TEACHING  OF  ARITHMETIC         III 

the  German  (and  particularly  the  Herbartian)  views 
of  the  bases  of  method,  or  the  basis  of  education, 
has  given  a  great  impetus  to  teaching  in  general, 
and  as  a  consequence  has  improved  the  teaching 
of  arithmetic.  For  the  application  of  these  views  to 
special  lessons  in  number  the  reader  is  referred  to 
the  works  of  these  writers.1 

The  whole  question  of  the  formal  steps  to  be  taken 
by  a  teacher  in  presenting  a  new  subject  to  a  class 
should  be  considered  apart  from  a  work  like  this.2 
Suffice  it  to  say  here  that  Rein,  whose  presentation 
of  the  matter  is  as  well  known  as  any,  sets  forth  five 
formal  steps  in  the  development  of  a  lesson :  i.  Prepa- 
ration ;  2.  Presentation ;  3.  Association ;  4.  Condensa- 
tion; 5.  Application.  Since  the  English  translations 
have  given  the  application  of  the  Herbart  method  to 
primary  work  only,  the  following  translation  of  a  fifth- 
grade  lesson  may  be  of  value. 

Aim.     How  shall  we  write  12  tenths  of  a  litre? 

i.   Preparation.    We  can  write  f  L,  J  L,  etc.     Instead 

1  De  Garmo,  Chas.,  The  Essentials  of  Method,  p.  117;  McMurry,  C.  A. 
and  F.  M.,  The  Method  of  the  Recitation,  p.  19.  For  the  best  working 
out  of  the  subject,  however,  one  must  consult  Rein,  Pickel  and  Scheller, 
Theorie  und  Praxis  des  Volksschulunterrichts,  6.  Aufl.,  Leipzig,  1898. 
A  brief  statement  of  the  application  of  the  formal  steps  to  elementary  \ 
arithmetic  is  given  in  Brautigam's  Methodik  des  Rechen-Unterrichts, 
2.  Aufl.,  Wien,  1895,  P-  *6>  and  in  several  other  similar  works. 

*  The  matter  is  clearly  presented,  historically  and  with  comparative 
tables,  in  De  Garmo's  Herbart,  New  York,  1896,  Chap.  V. 


112    THE  TEACHING  OF  ELEMENTARY   MATHEMATICS 

of  |-1.  we  can  also  write  ij  1. ;  instead  of  f  1.,  i-|l., 
etc.  In  what  other  way  can  we  write  yf  1.  ?  (iy%l.) 
Also  Tf  1.  ? 

2.  Presentation  of  the  new.     -J-J  or  I^-Q  can  also  be 
written  another  way.     We  already  know  that  -f-$  can 
be   written   0.2.       Further   examples.       What   does   a 
figure  before  the  decimal  point  indicate?     One  after 
the  decimal  point? 

3.  Association.     Compare  the  way  of  writing  i^  1. 
and   i.i  1. ;    3-^7 1.  and   3.3!.     Compare   i-J  1.   and  1.2!. 
Can  we  write  i\\.  as  we  write  i-j^l.  ? 

4.  Condensation.     If   we   have   to   write   more  than 
9  tenths  of  a  litre  we  reduce  the  tenths  of  a  litre  to 
whole   litres,  or  to  wholes  and  tenths,  and  we  place 
a  decimal  point  between  the  wholes   and   the  tenths 
(or  before  the  tenths,  or  after  the  wholes).     A  fourth 
or   an   eighth   of   a  litre   we   cannot   write   as   tenths. 
The  figures  after  the  dot  always  indicate  tenths. 

6.  Application.  Read  0.4;  0.6.  Read,  as  mixed 
numbers,  2.3  ;  4.6.  Reduce  to  tenths  2.3 ;  4.6.  Write 
24  wholes  and  7  tenths.  Write,  as  a  mixed  number, 
22  tenths.  Read,  as  tenths,  1.2;  2.3.1 

The  writing  of  numbers  —  Since  Pestalozzi's  time 
there  has  been  a  controversy  among  teachers  as  to 
whether  a  child  should  be  taught  the  Hindu  numerals 
along  with  the  numbers  themselves.  Pestalozzi,  as 
we  have  seen,  postponed  this  writing  until  the  child 

1  Rein,  Pickel  and  Scheller,  Theorie  und  Praxis,  V,  p.  237. 


THE  PRESENT  TEACHING  OF  ARITHMETIC         113 

had  a  knowledge  of  the  first  decade.  His  argument, 
the  limit  sometimes  being  changed  to  five,  meets  with 
much  approval  among  some  of  our  best  educators 
to-day.  Many  even  go  so  far  as  to  use  the  common 
symbols  of  operation  and  relation  before  the  Hindu 
numerals  are  learned,  giving  forms  like 


•• 


+•=•       1111111-111=1111 

II  in  HUH  =  ||| 


Others  ask,  and  with  reason,  why  a  symbol  like 
x  should  be  used,  but  not  one  like  4.  Still  others 
say,  also  with  much  reason,  that  the  common  psy- 
chological law  of  association  is  ample  warrant  for 
placing  before  the  child,  simultaneously,  the  forms 

III!          Four          4 

so  he  may  see  the  "one-to-one  correspondence,"  and 
fix  the  idea,  the  name,  and  the  symbol  together. 
This  view  is  taken  by  Hentschel,  one  of  the  leading 
German  writers  upon  method  in  arithmetic.  "The 
pupils,"  he  says,  "  have  now  seen  the  individual  num- 
bers represented  in  three  ways,  and  have  so  repre- 
sented them  for  themselves,  namely,  (i)  by  rows  of 
marks,  points,  etc.,  (2)  by  number  pictures,  and  (3)  by 
figures.  There  now  arises  the  question  as  to  which 
of  these  three  forms  shall  be  used  by  the  little  ones 
in  their  first  computations.  Can  we  at  once  put 


114    THE  TEACHING  OF  ELEMENTARY  MATHEMATICS 

them  into  work  with  the  figures?  For  myself  I  an- 
swer, yes." 1 

The  question,  as  is  usually  the  case  with  these 
disputed  matters  of  detail,  is  of  relatively  little  im- 
portance. The  experience  of  a  century  has  left  it 
entirely  unsettled,  the  results  being,  so  far  as  inves- 
tigations have  shown  as  yet,  quite  as  good  in  one 
case  as  the  other.  It  is  easy  to  theorize  upon  such 
a  point,  but  it  may  be  worth  while  to  consider  the 
difficulty  which  children  have  in  connecting  the  num- 
ber itself  with  the  proper  symbol  and  especially  with 
the  proper  name  in  the  number  series,  and  hence  to 
make  as  much  use  as  possible  of  the  law  of  associa- 
tion involved  in  presenting  the  number  picture,  the 
name,  and  the  symbol  simultaneously. 

The  work  of  the  first  year  —  The  majority  of  lead- 
ing writers  upon  the  subject  limit  the  results  of 
operations  to  the  number  space  i-io.  Some  go  to 
12.  Others  take  the  space  1-20,  and  the  argument 
is  a  strong  one  that  the  foundation  of  all  number 
work  lies  in  the  mastery  of  the  subject  in  this  space.2 
Many  advocate  counting  by  tens  during  the  second 
part  of  the  year,  and  then  filling  in  the  series,  thus 


1  Klotzsch,  Hentschel's  Lehrbuch   des   Rechenunterrichts  in  Volks- 
schulen,  14.  Aufl.,  Leipzig,  1891,  p.  10. 

2  E.g.  Grass,  J.,  Die  Veranschaulichung  beim  grundlegenden  Rechnen, 
Munchen,  1896.    This  work   gives  a  brief  but  valuable  resume  of  the 
leading  theories  of  first  grade  work. 


THE  PRESENT  TEACHING  OF  ARITHMETIC         115 

giving  the  child  a  number  space  beyond  that  in  which 
he  is  actively  working.  Such  a  plan  adds  to  the 
child's  interest,  and  allows  him  to  teach  himself  by 
the  talk  of  the  home.  On  the  whole,  present  experi- 
ence seems  to  show  that  the  number  space  1-20  for 
operations,  with  counting  forward  and  backward  in 
the  space  i-ioo  as  recommended  by  Tanck,  Knillmg, 
and  others,  forms  the  limit  of  the  working  curriculum 
of  the  first  year.  Whether  this  limit  can  be  reached 
depends  entirely  upon  the  class  of  pupils  and  the 
ability  of  the  teacher.  But  to  attempt  to  confine  not 
only  the  results  of  operations,  but  also  all  ideas  of 
number  to  the  space  i-io,  for  the  whole  year,  is 
not  only  unnecessary,  but  it  is  stupid  and  tedious 
for  the  children. 

The  great  desideratum  in  the  first  year's  work  is 
facility  in  handling  numbers,  not  in  solving  applied 
problems.  "Tell  me  a  story  about  four,"  is  harmless 
enough  at  first,  although  there  is  no  "story"  told; 
but  it  gets  to  be  a  very  old  story  before  the  year  is 
done.  Children  like  rapid  work  in  pure  number;  one 
has  but  to  step  into  a  class  whose  teacher  is  awake 
to  this  idea,  to  realize  the  fact ;  and  to  dawdle  through 
the  year  with  nothing  but  "story"  telling  about  num- 
ber not  only  leaves  ungratified  a  natural  desire,  but  it 
sows  the  seed  of  poor  number  work  thereafter.  There 
has  nothing  appeared  in  America  for  the  last  few  years 
that,  considering  its  brevity,  has  done  so  much  for  the 


Il6    THE  TEACHING  OF  ELEMENTARY  MATHEMATICS 

better  teaching  of  the  subject  as  President  Walker's 
little  monograph  on  "  Arithmetic  in  Primary  and  Gram- 
mar Schools."  1  He  cared  little  for  theories  and  meth- 
ods, but  he  went  to  the  root  of  the  subject  in  a  number 
of  his  observations.  "At  the  present  time  the  results 
in  accuracy,  if  not  in  facility,  of  arithmetical  work  leave 
very  much  to  be  desired.  Scarcely  has  the  child  been 
taught  to  count  as  high  as  ten,  when  he  is  put  at 
technical  applications  of  arithmetic,  to  money  coins,  to 
divisions  of  time,  space,  etc. ;  and  these  technical  appli- 
cations are  increased  in  number  and  in  difficulty  through 
the  successive  years  of  the  grammar  school,  until  for  a 
large  amount  of  so-called  arithmetic  the  pupil  gets  com- 
paratively little  practice  in  the  art  of  numbers."2  This 
must  not,  of  course,  be  construed  to  mean  that  the  child 
is  to  have  no  applied  arithmetic ;  it  is  simply  a  protest 
against  the  neglect  of  that  thorough  drill  in  pure  num- 
ber necessary  to  make  a  good  calculator. 

The  time  for  beginning  the  study  of  arithmetic  is  at 
present  a  matter  of  dispute.  Should  the  first  year  of 
the  subject,  above  mentioned,  be  synchronous  with  the 
first  school  year  ?  The  "  Committee  of  Fifteen  "  think 
not,  and  they  recommend  beginning  with  the  second 
school  year.  Before  Pestalozzi,  as  already  said,  the 
subject  was  not  begun  until  the  child  could  read.  Pes- 
talozzi, however,  recognized  that  the  child  has  as  much 
taste  for  numbers  as  for  letters,  and  proceeded  to  gratify 

i  Boston,  1887,  2  P.  II. 


THE  PRESENT  TEACHING  OF  ARITHMETIC          117 

this  taste  in  the  first  school  year,  a  plan  which  has  gen- 
erally been  followed  since  his  time.  This  idea  of  post- 
poning the  formal  study  of  number  until  the  second 
year  is  one  of  several  pre-Pestalozzian  ideas  which  have 
recently  appeared,  and  it  has  not  as  yet  impressed 
itself  upon  educators  as  one  of  great  importance.  That 
the  practical  results  for  arithmetic,  if  the  child  con- 
tinues to  the  seventh  grade,  will  probably  be  equally 
good,  is  true.  That  the  child  might  put  his  twenty 
w  minutes  a  day,  now  devoted  to  arithmetic,  to  better 
use,  may  be  true;  but  that  he  would  do  so  is  improb- 
able. Until  we  systematize  play,  and  put  the  time 
gained  from  primary  number  to  physical  exercise,  in 
the  open  air,  under  a  skilled  teacher,  it  is  doubtful  if 
the  child  should  give  up  the  few  minutes  a  day  in  a 
line  of  work  for  which  he  has  a  taste  and  about  which 
he  delights  to  know. 

Oral  arithmetic  —  The  oral  arithmetic,  so  necessary 
before  the  Hindu  numerals  made  written  computation 
easy,  fell,  as  we  have  seen,  into  disfavor  at  the  Renais- 
sance. Revived  by  Pestalozzi  and  his  contemporaries, 
it  had  much  favor  not  only  in  Europe,  but  also,  thanks 
to  Colburn's  excellent  work,  in  America.  But  the 
advent  of  cheap  slates  and  paper  and  pencils  seems 
to  have  driven  it  out  of  our  schools  for  a  generation. 
It  is  now  reviving,  and  it  is  to  be  hoped  that  we 
shall  not  again  cease  to  secure  reasonable  facility  in 
rapid  oral  work  with  the  ordinary  numbers  of  daily  life, 


Il8     THE  TEACHING  OF  ELEMENTARY  MATHEMATICS 

The  subject  can  easily  be  carried  to  an  extreme;  but 
within  reasonable  limits  it  should  be  demanded  in  every 
grade.  It  lubricates  the  arithmetical  machine,  and  five 
minutes  a  day  to  this  subject  could  hardly  fail  to  bring 
all  pupils  to  reasonable  facility  with  numbers. 

Treating  the  processes  simultaneously  —  This  is,  of 
course,  as  impossible  as  it  is  to  have  several  bodies 
occupy  the  same  space  at  the  same  time.  But  the 
expression  means  the  so-called  mastery  of  a  number, 
the  study  of  the  four  processes,  before  the  next  is 
studied.  As  already  stated,  this  is  the  essence  of  the 
Grube  method,  its  fundamental  feature  as  well  as  its 
fundamental  defect.  "  It  seems  absurd,  or  worse  than 
absurd,  to  insist  on  thoroughness,  on  perfect  number 
concepts,  at  a  time  when  perfection  is  impossible  .... 
If  the  child  knows  three,  if  he  has  even  an  intelligent 
working  conception  of  three,  he  can  proceed  in  a  few 
lessons  to  the  number  ten,  and  will  thus  have  all  higher 
numbers  within  comparatively  easy  reach."1  A  more 
tedious  way  of  presenting  number  than  that  of  Grube' s 
would  be  hard  to  find,  and  yet,  in  America  and  Ger- 
many, this  feature  still  has  a  considerable  following. 

The  spiral  method  —  In  the  preparation  of  text- 
books we  have  had  various  experiments  of  late,  all  the 
result  of  the  restless  desire  to  break  away  from  the 
bad  features  of  the  older  works.  The  so-called  "  spiral 
method"  seems  to  have  been  first  suggested  by  Ruh- 

1  McLellan  and  Dewey,  The  Psychology  of  Number,  p.  172,  176. 


THE  PRESENT  TEACHING  OF  ARITHMETIC         119 

sam,1  and  to  have  found  little  favor  anywhere  until  it 
was  recently  taken  up  in  America.  It  consists  in 
taking  the  class  around  a  circle,  say  with  the  topics  of 
common  fractions,  decimal  fractions,  greatest  common 
divisor,  and  square  root;  then  swinging  around  again 
on  a  broader  spiral,  taking  the  same  topics,  but  with 
more  difficult  problems;  then  again,  and  so  on  until 
the  subjects  are  sufficiently  mastered. 

The  idea  has  much  to  recommend  it.  A  child  is 
not  now  expected  to  master  common  fractions  by  going 
once  over  the  subject  and  then  leaving  it  forever.  And 
yet  the  older  text-books  expected  him  to  do  just  that 
for  greatest  common  divisor,  square  root,  etc.  But  the 
idea  can  easily  be  carried  to  an  extreme,  the  class 
swinging  around  the  spirals  so  frequently  as  to  pro- 
duce mathematical  nausea.  It  is  a  question  how 
elaborate  the  scheme  should  be  made,  and  it  has  not 
been  sufficiently  tried  to  answer  this  question. 

Common  vs.  decimal  fractions — The  question  of 
sequence  of  common  and  decimal  fractions  is  one 
which  has  recently  been  much  discussed.  It  is  easy 
to  dismiss  the  whole  subject  by  some  such  remark  as, 
"Logically  the  decimal  fraction  comes  first,  because  it 
grows  naturally  out  of  our  number  system,"  and  this 
is  frequently  done  in  some  educational  sheets.  Another 

1  Aufgaben  fur  das  praktischen  Rechnen  zum  Gebrauch  in  den  un- 
tern  drei  Klassen  der  Realschulen  und  in  den  obern  Klassen  von  Bur* 
gerschulcn  in  drei  concentrisch  sich  erweiternden  Cursen,  1866. 


I2O    THE  TEACHING  OF  ELEMENTARY   MATHEMATICS 

will  say  that  the  Prussian  educational  decree  of  1872 
put  the  decimal  fractions  first,  and  that  the  experience 
of  these  many  years  has  proved  the  wisdom  of  the 
plan.  But  just  as  strong  an  argument  can  be  advanced 
by  saying  that  psychologically  the  common  fraction 
should  precede,  because  the  concept  is  the  simpler; 
that  historically  it  was  in  use  long  before  the  decimal 
system  of  writing  numbers  was  known,  to  say  nothing 
of  the  decimal  fraction;  and  that  Prussia's  experiment 
has  been  productive  of  such  doubtful  results  that  Baden, 
and  Bavaria,  and  Saxony  still  follow  the  older  plan.1 

The  question  is  really,  however,  one  belonging  rather 
to  the  old-fashioned  course  than  to  the  modern,  to  the 
days  when  the  pupil  was  expected  to  "master"  com- 
mon fractions  before  studying  the  decimal.  Our 
modern  arithmetics,  of  any  standing,  follow  no  such 
plan.  The  fact  is,  no  one  ever  thinks,  practically,  of 
teaching  0.5  before  J,  or  0.25  before  ^.  The  simple 
fractions  ^,  ^,  enter  into  the  work  of  the  first  year;  the 
forms  0.5,  0.25,  represent  a  much  greater  degree  of  ab- 
straction, and  hence  should  have  place  considerably  later. 

But  on  the  other  hand,  as  between  adding  0.5  and 
0.25,  or  |~||-  and  |-|-£,  there  can  be  no  question  as  to 
which  should  have  first  place.  And  hence  the  con- 
clusion will  probably  be  reached  by  most  teachers  that 

1  For  details  as  to  these  state  systems  see  Dressier,  Der  mathe- 
matisch-naturwissenschaftliche  Unterricht  an  deutschen  (Volksschullehrer-) 
Seminaren,  Hoffmann's  Zeitschrift,  XXIII.  Jahrg.,  p.  15. 


THE  PRESENT  TEACHING  OF  ARITHMETIC          121 

the  elementary  treatment  of  simple  fractions  has  the 
first  place,  but  that,  long  before  the  pupil  comes  to  the 
serious  difficulties  of  the  common  fraction,  the  tables 
of  United  States  money,  or  possibly  those  of  the  metric 
system,  should  make  him  familiar  with  the  decimal 
forms  and  the  simple  operations  therewith. 

Improvements  in  algorism,  that  is,  in  the  arrangement 
of  work  in  performing  the  elementary  operations,  are 
constantly  appearing,  and  some  are  of  real  value.  Two 
which  are  now  struggling  for  acceptance,  with  every 
prospect  of  success,  may  be  mentioned  here  as  types. 

In   subtracting  297  from  546,  we  have  the     -  g 
two  old  plans,  both  dating  from  the  time  of     2Q7 
the  earliest  printed   text-books,  at  least.     The 
calculation  is  substantially  this: 

1.  7  from  1 6,  9;  9  from  13,  4;   2  from  4,  2;   or 

2.  7  from  16,  9;  10  from  14,  4;  3  from  5,  2. 
But  we  have  also  a  more  recent  plan : 

3.  7  and  9,  16;  10  and  4,   14;  3  and  2,  5. 

To  this  might  be  added  a  fourth  plan  which  has 
some  advocates : 

4.  7  from  10,  3 ;  3  and  6,  9 ;  9  from  10,  i ;  I  and  3, 
4;  2  from  4,  2. 

All  four  of  these  plans  are  easily  explained,  the 
first  rather  more  easily  than  the  others.  But  the 
third  has  the  great  advantage  of  using  only  the  addi- 
tion table  in  both  addition  and  subtraction,  and  of 
saving  much  time  in  the  operation.  It  is  the  so- 


122    THE  TEACHING  OF  ELEMENTARY  MATHEMATICS 

called  "  Austrian  method  "  of  subtraction.  The  fourth 
plan,  while  a  very  old  one  and  possessed  of  some 
good  features,  is  so  ill  adapted  to  practical  work  as 
to  have  no  place  in  the  school.  It  is  hardly  neces- 
sary to  say  that  the  old  expressions,  "borrow"  and 
"carry,"  in  subtraction  and  addition  are  rapidly  going 
out  of  use  ;  they  were  necessary  in  the  old  days  of 
arbitrary  rules,  but  they  have  no  advocates  of  any 
prominence  to-day. 

In  division  we  have  also  an  "Austrian  method,"  a 
valuable  arrangement.  It  is  not  long  since  a  prob- 
lem like  6.275-^-2.5  was  "worked"  by  a  rule  which 
was  rarely  developed.  Now  the  work  is  arranged  in 
this  way  : 

2.51 
2.5)6.275  25)62.75 

50 


12.5 
0.25 
0.25 

Such  an  arrangement  leaves  no  trouble  with  the 
decimal  point,  and  the  work  is  easily  explained.  In 
the  above  problem  the  entire  remainder  is  brought 
down,  and  the  decimal  point  is  preserved  throughout, 
as  should  be  done  until  the  process  is  thoroughly 
understood  ;  then  the  abridgment  should  appear. 


THE  PRESENT  TEACHING  OF  ARITHMETIC         123 

The  explanations  of  greatest  common  divisor,  divi- 
sion of  fractions,  etc.,  are  so  fully  given  in  any  of 
our  recent  American  text-books  that  it  is  not  worth 
while  to  attempt  them  in  a  work  of  this  nature. 

The  formal  solution  of  applied  problems  is  now 
generally  recognized  as  logic  work  as  well  as  number 
work.  The  result  of  the  problem  is  as  important  as 
ever,  but  it  is  not  all-important ;  the  value  of  a  logi- 
cal explanation  is  now  recognized  —  of  course  when 
the  pupil  has  reached  the  proper  grade.  Hence  the 
solutions  of  problems  in  percentage  and  in  analysis 
are  now  generally  given  in  step  form,  the  actual 
work  of  the  elementary  operations  being  omitted. 
For  example: 

A  commission  merchant  remits  $1073.50  as  the  net 
proceeds  of  a  sale  after  deducting  5%  commission; 
required  the  amount  received  from  the  sale. 

1.  0.95  of  the  amount  =  $1073.50. 

2.  /.  the  amount  =  $1073.50  •*•  0.95  =  $1130, 

by  dividing  these  equals  by  0.95. 

Or  better  still,  by  letting  x  represent  this  amount 
(not  the  number  of  dollars,  since  we  are  preserving 
the  dollar  sign  before  the  other  numbers), 

1.  0.95*  =  $1073.50 

2.  .*.     *=  $1073. 50-*- 0.95 

=  $1130. 


124    THE  TEACHING  OF  ELEMENTARY   MATHEMATICS 

This  introduces  the  equation  form  in  a  more  pro- 
nounced way,  but  this  is  now  generally  approved  by 
educators.1 

There  are  still  some  advocates  of  the  following 
plan: 

1.  95%  of  the  amount  is  $1073.50. 

2.  /.       i%  of  the  amount  is  ^  of  $1073.50=  $11.30. 

3.  /.  100%  of  the  amount  is  100  x  $11.30  =  $1130. 

This,  the  unitary  method,  is  by  some  thought  to 
be  simpler  than  the  others,  though  why  it  is  simpler 
to  derive  o.oi  from  0.95  than  to  derive  I  from  0.95, 
it  is  difficult  to  say. 

The  following  form  has  also  an  occasional  advo- 
cate: 

1.  Let  1 00%  equal  the  amount. 

2.  Then  100%  —  5%  =  95%. 

3.  If       95%  =  $1073. 50, 
4-  i%  =  $     11-30, 
5.  and  100%  =  $1130. 

This  is  a  relic  of  the  mediaeval  method  of  "  false  posi- 
tion," a  pre-algebraic  device.  The  100%  is  merely  I, 
and  we  begin  by  letting  this  i  equal  the  unknown 

1"Alle  Padagogen  sind  hierin  einverstanden."  Hentschel,  p.  81. 
"  Can  any  one  imagine  a  good  teacher,  who  is  also  a  good  algebraist,  who 
will  not  train  his  pupils  to  use  letters  for  numbers  long  before  arithmetic 
is  completed?"  Safford,  T.  H.,  Mathematical  Teaching,  Boston,  1887, 
p.  23.  The  question  is  discussed  in  a  broad  way  by  Schuster,  M.,  Die 
Gleichung  in  der  Schule,  in  Hoffmann's  Zeitschrift,  XXIX.  Jahrg.,  p.  81. 


THE  PRESENT  TEACHING  OF  ARITHMETIC          125 

quantity.  Of  course  x  or  any  other  symbol  might  be 
used  to  better  advantage,  for  we  know  very  well  that 
the  unknown  quantity  is  not  i.  Furthermore,  95% 
does  not  equal  $1073.50;  it  is  95%  of  the  amount ', 
or  of  x,  that  equals  $1073.50. 

By  following  such  a  plan  as  the  one  first  mentioned 
the  well-founded  complaint  against  the  thoughtless 
mechanism  of  the  past  disappears.  Instead  of  words 
and  rules  without  content,  there  is  content  with  a 
minimum  of  words  and  with  no  unexplained  rule.1 

It  is  only  a  few  years  back  that  such  forms  as 
"2  ft.  x  3  ft.  =  6  sq.  ft,"  "2x3  =  6  ft.,"  "  24  cu.  ft. 
-*-  8  sq.  ft.  =  3  ft,"  and  the  like  were  not  uncommon. 
Now,  however,  all  careful  teachers  are  insisting  that 
such  inaccuracies  of  statement  beget  inaccuracy  of 
thought  and  hence  should  not  be  tolerated  in  the 
schoolroom.  It  is  true  that  these  all  depend  upon 
the  definitions  assumed,  and  that  well-known  teachers 
have  advocated  such  a  change  of  definition  as  will 
allow  of  saying  "4  ft.  x  2  yds.  =  3456  sq.  in."3;  but, 


1  Die  Kinder  .  .  .  losen  einschlagige  Aufgaben,  aber  alles  das  geschieht 
meistens  auf  mechanischem  Wege.     Wir  finden  Worte  und  Regeln  ohne 
Inhalt.     Fitzga,  p.  5.    The  other  side  of  the  case,  the  danger  of  using 
algebra  unnecessarily  is  presented   in  Supt.  Greenwood's  Dissent  from 
Dr.  Harris's  Report  of  the  Committee  of  Fifteen. 

2  This  illustration,  from  an  article  by  Professor  A.  Lodge  in  the  General 
Report  of  the  Association  for  the  Improvement  of  Geometrical  Teaching, 
January,  1888.     Similar  articles  have  appeared  in  Hoffmann's  Zeitschrift 
in  recent  years. 


126    THE  TEACHING   OF  ELEMENTARY  MATHEMATICS 

with  our  present  definitions,  such  forms  lead  to  great 
looseness  of  thought. 

It  is  the  loose  manner  of  writing  out  solutions,  tol- 
erated by  many  teachers,  that  gives  rise  to  half  the 
mistakes  in  reasoning  which  vitiate  pupils'  work.  The 
carelessness  in  form  begets  that  carelessness  of  thought 
which  gives  point  to  such  amusing  absurdities  as  these : 

1.  A  bottle  £  full  =  a  bottle  |  empty.     Divide  by  £, 

.*.  a  bottle  full  =  a  bottle  empty. 

2.  20  dimes  =  2  dollars.     Square  each  member  and 
.*.  400  dimes  =  4  dollars.1 

Longitude  and  time  furnish  a  type  of  the  applied 
problems  of  arithmetic,  one  in  which  much  careless- 
ness of  form  and  thought  is  often  apparent,  and  as 
such  it  is  entitled  to  some  special  consideration. 

The  subject  is  best  presented,  perhaps,  by  a  brief 
discussion  of  the  question  of  the  relative  positions  of 
the  sun  and  earth  at  the  hour  of  the  class  recitation, 
the  globe  being  held  before  the  class,  the  northern 
hemisphere  visible,  and  North  America  being  on  the 
lower  half  so  as  to  be  recognized  easily  (it  being 
then  "  right  side  up  "  to  the  pupils).  The  sun  being 
located,  the  question  of  the  forenoon  and  the  after- 
noon on  the  earth's  surface  may  be  discussed,  then 
the  position  of  midnight,  then  the  effect  of  the  revo- 
lution of  the  earth  with  respect  to  these  periods ;  and 

1  Adapted  from  Rebiere,  A.,  Mathematiques  et  mathematicians,  2.  ed., 
Paris,  1893,  p.  331. 


THE  PRESENT  TEACHING  OF  ARITHMETIC         127 

finally,  for  one  lesson,  the  number  of  degrees  through 
which  the  schoolhouse  and  vicinity  must  pass  in  order 
that  the  time  shall  be  24  hours  later. 

All  this  leads  to  the  development  of  two  tables, 
the  foundations  upon  which  the  subject  rests: 

TABLE  I 

360°  correspond   to  24  hrs. 

.-.  i°  corresponds  to  3^  of  24  hrs.  =  ^  hr.  =  4  min. 
.*.  if  corresponds  to  ^  of  4  min.  =  -^  min.=  4  sec. 
/.  i"  corresponds  to  -fa  of  4  sec.  =  -^  sec. 

TABLE  II 

24  hrs.  correspond    to  360°. 
/.  I  hr.     corresponds  to  ^¥  of  360°  =  15°. 
.-.  i  min.  corresponds  to  -fa  of    15°  =  £  of  i°  =  15'. 
/.  i  sec.  corresponds  to  ^  of    15'    =  £  of  i'  =  15". 

To  say  that  360°  =  24  hrs.  is  as  inaccurate  as  to 
say  that  $  4  =  24  Ibs.  of  beef ;  there  may  be  some 
correspondence,  as  in  value,  etc.,  but  there  is  no 
such  equality  as  is  set  forth  in  the  statement. 

The  theory  of  the  subject  is  now  best  brought  out 
by  numerous  simple  oral  problems  of  this  nature:  If 
the  difference  in  longitude  between  two  ships  is  10°, 
what  is  their  difference  in  time?  If  their  difference 
in  time  is  20  min.,  what  is  their  difference  in  longi- 
tude? To  make  such  problems  practical,  cases  of 


128     THE  TEACHING  OF   ELEMENTARY  MATHEMATICS 

ships  or  observatories  should  be  used,  since  the  recent 
rapid  development  of  standard  time  has  shut  out  local 
time  in  the  large  majority  of  places  in  the  civilized  world. 

Written  solutions .  may  now  be  required  in  some 
such  form  as  the  following : 

The  difference  in  longitude  between  two  ships  is 
10°  45'  30",  required  the  difference  in  time. 

1.  10  x  4    min.  =  40  min. 

2.  45  x  -j1^  min.  =  3  min.  (or  45  x  4  sec.  =  180  sec. 
=  3  min.). 

3.  30  x  TV  sec.  =  2  sec.  4.   /.  43  min.  2  sec. 

The  difference  in  time  between  two  ships  is  43 
min.  2  sec.,  required  the  difference  in  longitude. 

1.  43  x  I  of  i°  =  iof°  =  10°  45'  (or  43  x  15'=  •••) 

2.  2x15"  =  30"-  3-  •*•  10°  45'  30". 

Some  of  the  older  arithmetics  still  write  "2  hr. 
3'  15""  for  2  hr.  3  min.  15  sec.,  or  2  h.  3  m.  15  s., 
but  it  is  unwise  to  change  the  general  custom  of 
using  the  '  and  "  for  longitude  only.  More  serious 
is  their  adherence  to  the  mechanical  rule,  and  to 

such  forms  as  these: 

43  mm.     2  sec. 

15  |io°     45'         30"  15 

|  hr.  3  min.  2  sec.  645'          30" 

=  43  min.  2  sec.  =  10°  45'  30" 

Explain  all  we  will,  such  forms  tell  the  eye  that 
degrees  divided  by  an  abstract  number  give  hours, 


THE  PRESENT  TEACHING  OF  ARITHMETIC         129 

and  that  time  is  transformed  by  some  miracle  into 
longitude  by  multiplying  by  15!  Text-book  makers 
may  argue  for  brevity,  but  the  astronomer  and  the 
navigator  who  wish  brevity  always  use  longitude 
tables.  It  is  not  brevity  that  we  seek;  it  is  an 
understanding  of  the  process. 

The  two  points  at  which  the  teacher  needs  to  aim, 
after  the  elementary  correspondence  between  longi- 
tude and  time  is  fixed,  are  (i)  standard  time,  and 
(2)  the  date  line.  The  old-style  complicated  prob- 
lems may  well  give  way  to  these  new  and  interesting 
topics.  The  last  decade  of  the  nineteenth  century 
has  seen  standard  time  made  well-nigh  universal  in  the 
highly  civilized  portions  of  the  world,  and  the  recent 
events  in  the  Philippines  have  given  to  the  subject  of  the 
date  line  even  greater  interest  for  American  pupils.1 

Ratio  and  proportion  still  maintain  their  conven- 
tional copartnership  in  most  of  our  arithmetics, 
usually  setting  forth  an  array  of  problems  inherited 
from  some  generations  past.  There  is  just  now  a 
good  deal  said  about  introducing  the  ratio  concept 
earlier  in  the  course,  and  this  may  happily  break  up 
the  partnership  and  show  ratio  as  the  important  sub- 
ject which  it  really  is. 

At    present,    in    the    standard    type    of    arithmetic, 

1  For  a  full  discussion  of  these  two  subjects,  with  late  information  con- 
cerning standard  time,  and  with  maps  showing  the  date  line,  the  reader  is 
referred  to  Be  man  and  Smith's  Higher  Arithmetic,  Boston,  1897. 
K 


130    THE  TEACHING  OF  ELEMENTARY   MATHEMATICS 

ratio  has  place  merely  as  an  introduction  to  pro- 
portion. The  latter  subject  is  taught  as  a  matter 
of  rule,  as  if  it  were  to  be  used  so  often  as  to 
justify  this  unscientific  treatment.  The  fact  is,  the 
subject  is  rarely  used  in  business,  and  almost  its 
only  arithmetical  applications  of  value  are  to  be 
found  in  physical  problems  and  in  problems  involving 
similar  figures.  Before  simple  equations  were  invented 
the  subject  had  much  more  value  than  at  present, 
and  the  arbitrary  "  Rule  of  Three,"  as  it  was  called, 
may  have  been  justifiable.  At  present,  to  teach  the 
subject  by  mere  rule,  or  by  any  such  senseless  device 
as  the  "cause  and  effect"  method,  is  unwarranted. 

There  is  just  now  a  growing  reform  in  presenting 
proportion.  This  movement  employs  the  fractional 
notation,  with  which  the  pupil  is  familiar,  and  the 

common  equation  form,  thus  :  -  =  -^,  to  find  x.     Mul- 

O 

tiplying  these  equals  by  3,  x  —  f . 

Consider,  for  example,  a  single  applied  problem : 
If  a  plumb  line  i  yd.  long  casts  a  shadow  6  ft.  long, 
how  high  is  an  adjacent  flagstaff  which  at  the  same 
instant  casts  a  shadow  84  ft.  long  ? 

i.  Let  x  —  the  number  of  feet  required. 
Then  —  or  -  =  the  ratio  of  the  heights, 

and       -4 — '  or  -4  =  the  ratio  of  the  shadow  lengths, 
oft.         6 


THE  PRESENT  TEACHING  OF  ARITHMETIC          131 

2.  And   since   the   heights    are   proportional  to   the 

shadow  lengths, 

x     84 

—  *"**  i     _", , 

3      6 

3.  Multiplying  by  3,  x  =  42. 
.-.  the  staff  is  42  ft.  high. 

After  the  class  is  familiar  with  the  theory,  the  work 
should  be  given  with  the  other  symbols,  because 
these  are  needed  in  common  scientific  reading,  thus : 
x :  3  =  84  :  6,  or  even  the  antiquated  form  x :  3  : :  84 :  6. 

Solutions  of  this  nature,  with  the  reasoning  set 
forth,  give  us  the  "  thought  reckoning  "  (Denkrechnen) 
which  our  best  educators  demand,  in  place  of  the  rule- 
work  of  the  old  school.1 

Square  root  was  formerly  treated  geometrically, 
that  being  the  plan  inherited  from  the  Greeks,  the 
nation  which  most  excelled  in  geometry  in  ancient 
times.2  But  the  method  which  follows  the  algebraic 
formula  is  preferable  on  many  accounts.  The  fact  that 
the  square  on  f  +  n  is  f2  +  2  fn  -f-  n2,  where  f  stands 
for  the  found  part  of  the  root  and  n  for  the  next 
figure,  may  profitably  be  pictured  by  a  geometric 

1  The  general    question    of   proportion    is    discussed  in    a    valuable 
article  by  Dressier,  Der  mathematisch-naturwissenschaftliche  Unterricht 
an    deutschen    (Volksschullehrer-)    Seminaren,    Hoffmann's    Zeitschrift, 
XXIII.  Jahrg.,  I. 

2  Theon  of  Alexandria,  father  of  Hypatia,  gave  the  common  geomet- 
ric plan.    Gow,  History  of  Greek  Mathematics,  p.  55;  Cantor,  I,  p.  460. 


132    THE  TEACHING  OF  ELEMENTARY   MATHEMATICS 

diagram.     But  the  formula   is  to   be   preferred  to  the 
diagram,  as  a  basis  for  work,  because 

1.  The  geometric  notion  limits  the  idea  of  involution 
to  the  square  and  cube  roots; 

2.  The  formula  method  makes  the  cube  and  higher 
roots  very  simple  after  square  root  is  understood; 

3.  We  are  working  with  numbers,  not  with  geometric 
concepts ; 

4.  The  formula  lends  itself  more  easily  to  a  clear 
explanation  of  the  process. 

One  of  the  great  difficulties  in  explaining  square  root 
lies  in  the  fact  that  tradition  has  encumbered  it  with 
superfluous  difficulties.  Consider,  for  instance,  the 
question,  "Why  do  we  separate  into  periods  of  two 
figures  each,  beginning  at  the  right?"  The  answer 
might  be  given,  "We  need  not  do  so;  it  was  neces- 
sary when  square  root  was  merely  a  matter  of  rule; 
if  one  thinks,  such  separation  is  quite  unnecessary; 
furthermore,  we  would  not  begin  at  the  right  anyway, 
but  rather  at  the  decimal  point,  this  rule  having  been 
framed  long  before  the  decimal  point  was  known." 
Again,  "Why  do  we  bring  down  only  one  period  at  a 
time?"  For  reply  we  may  say,  "We  don't;  it  is  much 
better  for  beginners  to  bring  down  all  of  the  remainder 
each  time,  because  it  makes  the  explanation  easier." 
Of  course,  after  the  complete  process  is  fully  under- 
stood we  may  adopt  this  and  other  abridgments  if 
we  desire,  and  then  the  explanation  is  not  difficult; 


THE  PRESENT  TEACHING  OF  ARITHMETIC          133 

but  it  is  very  poor  policy  to  let  such  unnecessary 
questions  enter  at  a  time  when  the  teacher  is  seeking 
to  have  the  process  clearly  understood. 

It  may  be  said  that  these  suggestions  and  the  follow- 
ing solution  make  the  process  longer  than  necessary. 
But  since  almost  the  sole  justification  for  the  subject 
of  involution  is  the  fact  that  it  offers  training  in  logic, 
this  training  is  of  paramount  importance.  For  practical 
purposes  the  square  root  is  usually  extracted  by  the 
help  of  tables. 

A  problem  in  square  root  might,  then,  be  arranged 
as  follows: 

23.4   =  root 
547.56  contains  some  square,/2+2/#  +  u2 


2/=  40     147.56  contains  2/«  +  w2,  where  /=  20 
=  43      129      =2/«  +  «a 


2/=  46       18.56  contains  2  fn  +  «2,  where  /=  23 
2/+  n  =  46.4     18.56  =  2/«  +  «2 

This  arrangement  shows  what  each  number  equals 
(exactly  or  approximately),  and  the  only  things  to 
explain  are  (i)  these  equalities,  and  (2)  why  2  f  is 
taken  as  the  "trial  divisor,"  matters  offering  no  diffi- 
culties.1 

1  For  full  explanation,  and  for  other  suggestions  as  to  the  factoring 
method,  treatment  of  fractions,  the  double  sign,  etc.,  see  Beman  and 
Smith's  Higher  Arithmetic,  Boston,  1897,  p.  35. 


134    THE  TEACHING  OF  ELEMENTARY  MATHEMATICS 

The  metric  system  —  The  common  measures  of  daily 
life  demand  great  attention  in  arithmetic.  Until  they 
have  become  thoroughly  familiar,  until  they  have  taken 
prominent  place  in  the  child's  mind,  until  they  have 
been  taught  with  the  actual  measures  (as  far  as  may 
be)  in  hand,  and  until  they  have  been  practically  used 
in  hundreds  of  concrete  problems,  the  metric  system 
has  no  place.  The  child  can  get  along  for  a  while 
without  this  system;  indeed,  he  may  never  be  con- 
scious of  a  loss  if  he  does  not  know  it;  but  the  com- 
mon system  he  needs  daily. 

On  the  other  hand,  as  compared  with  the  apothecaries' 
and  troy  measures,  or  with  leagues,  furlongs,  barley- 
corns, pipes,  tuns,  quintals,  etc.,  the  metric  system 
should  certainly  have  precedence. 

Only  two  or  three  bits  of  advice  to  the  teacher  need 
be  given.  First,  these  measures,  like  all  others  taught 
to  the  child,  should  be  actually  in  hand ;  they  must  be 
made  to  seem  real  by  abundant  use;  merely  to  learn 
the  tables  is  of  little  value.  The  French  schools,  with 
their  little  cases  of  metric  units  on  the  front  wall  of  the 
recitation  rooms,  always  within  sight  of  the  children, 
set  an  example  worthy  of  our  attention.1 

Again,  the  child  will  probably  use  the  system  by 
itself  if  at  all;  that  is,  he  will  not  be  translating  back 
and  forth  with  the  common  system.  To  ask  how  many 
grammes  in  4  cwt.  37  Ibs.  2  oz.,  is  worthless  as  a  practical 

1  See  also  Fitzga,  I,  p.  41,  57. 


THE  PRESENT  TEACHING  OF  ARITHMETIC          135 

problem;  it  gives  the  child  a  little  "figuring,"  but  it 
destroys  his  appreciation  of  the  great  advantages  of 
the  modern  system.  A  few  of  the  common  units  may 
be  translated,  as  in  a  question  like  this :  A  traveller  in 
Germany  is  allowed  25  kilos  of  baggage  free;  about 
how  many  pounds  is  this  ?  But  such  translation  should 
be  confined  to  common  cases  and  to  oral  work. 

The  pupil  should  be  led  to  see  that  the  names  are 
not  so  strange  as  might  at  first  appear.  As  a  gas- 
metre  measures  gas,  and  a  water-metre  measures  water, 
so  a  metre  is  a  unit  of  measure  ;  it  is  a  little  longer  than 
our  yard.  And 

as  a  mill  is  o.ooi  of  $i,  so  a  millimetre  is  o.ooi  of 
I  metre; 

as  a  cent  is  o.oi  of  $i,  so  a  centimetre  is  o.oi  of 
I  metre; 

as  a  decimal  point  comes  before  tenths,  so  a  deci- 
metre is  o.i  of  i  metre; 

as  a  dekagon  is  a  lo-angled  figure,  so  a  dekametre 
is  10  metres. 

So  milli-  means  o.ooi,  deci-  means  o.i, 

centi-  means  o.oi,  deka-  means  10, 

and  there  are  only  three  new  prefixes  to  learn : 

hekto-,  which  means        100, 
kilo-,  which  means      1000, 
-t  which  means  10,000. 


136    THE  TEACHING  OF  ELEMENTARY  MATHEMATICS 

With  these  prefixes  well  in  mind  the  tables  of  the 
metric  system  are  practically  known.  Hence  a  great 
deal  of  the  oral  drill  in  this  work  may  profitably  be 
devoted  to  these  prefixes,  taking  them  at  random 
and  asking  their  numerical  equivalents,  and  vice 
versa. 

The  grade  in  which  the  metric  system  is  taught  is 
determined  largely  by  the  science  work  in  the  school. 
Since  all  science  now  uses  this  system,  it  may  be  taken 
up  as  soon  as  simple  physical  problems  are  introduced. 
But  reference  is  so  frequently  made  to  the  system  in 
the  current  literature  of  the  day,  that  to  postpone  the 
subject  beyond  the  eighth  grade,  or  to  teach  it  in  a 
perfunctory  manner,  is  unwarranted. 

The  applied  problems,  and  especially  the  business 
problems  involving  percentage,  are  so  well  adjusted 
to  the  uses  and  capacities  of  the  various  grades, 
in  the  modern  American  text-books,  that  little  need 
be  said  upon  the  subject.  But  topics  like  true  dis- 
count, equation  of  payments,  partnership,  involving 
time,  arbitrated  exchange,  insurance  as  it  was  fifty 
years  ago  —  these  subjects  have  no  place  in  the  com- 
mon school  arithmetic  of  to-day.  Our  recent  books 
generally  print  pictures  of  drafts,  checks,  notes,  etc., 
and  give  such  explanations  of  common  business  cus- 
toms as  render  these  intelligible  to  pupils  before  they 
leave  the  eighth  grade.  Such  helps,  and  the  study 
of  the  actual  documents  in  the  classroom,  will  si- 


THE  PRESENT  TEACHING  OF  ARITHMETIC         137 

lence  much  of  the  prevalent  criticism  that  we  teach 
too  much  for  the  school  and  too  little  for  life.1 

"Short  cuts"  —  The  short  methods  so  much  sought 
in  earlier  times  are  now  less  in  demand.  The  reason 
is  not  that  time  is  considered  less  precious,  but  that 
the  "  short  cuts  "  have  been  found  generally  to  apply 
to  problems  of  no  importance,  or  that  the  elaborate 
use  of  tables  has  rendered  them  unnecessary.  For 
example,  it  was  once  considered  a  mark  of  an  ex- 
pert accountant  to  have  at  hand  numerous  short 
methods  of  reckoning  interest;  now  the  accountant 
turns  at  once  to  his  interest  tables,  and  the  average 
man  with  no  tables  at  hand  has  forgotten  the  rules 
of  his  school  days. 

Formerly  the  expression  "75°  -s-  15  =  5  hrs."  was 
allowed  on  the  score  that  its  brevity  justified  its 
falsity;  now,  any  one  who  has  occasion  to  solve  prob- 
lems of  this  kind  in  a  practical  way  resorts  to  tables. 
Formerly,  mere  rule  work  was  justified  in  square  and 
cube  root  on  the  plea  of  brevity ;  now,  for  practical 
purposes,  we  generally  extract  such  roots  by  loga- 
rithmic or  evolution  tables. 

Mensuration  was  formerly  taught  solely  by  rule. 
Even  now  the  strictly  scientific  treatment  belongs  to 
geometry.  But  there  are  certain  propositions  that  are 
so  commonly  needed  that  they  must  have  place  in 
arithmetic  for  those  who  may  not  study  geometry. 

1  Viclfach  nur  fiir  die  Scbule  und  nicht  fur  das  Leben.    Fitzga,  I,  p.  6. 


138    THE  TEACHING  OF  ELEMENTARY  MATHEMATICS 

Such  are  the  propositions  which  give  the  formulae 
for  measuring  the  square,  or  more  generally  the  rec- 
tangle and  the  parallelogram,  the  triangle,  possibly  the 
trapezoid,  the  circle,  the  parallelepiped,  the  cylinder, 
and  possibly  also  the  cone  and  sphere. 

The  mensuration  of  these  figures  may  easily  be 
taken  up  in  arithmetic  in  a  reasonably  scientific  way, 
and  this  is  outlined  in  most  of  our  modern  text- 
books. For  example,  the  computation  of  the  area 
of  a  rectangle  2  in.  by  3  in.  is  easily  made  a  matter 
of  reason  by  using  a  figure  illustrating  the  statement 
2  x  3  x  i  sq.  in.  =  6  sq.  in.,  or  the  statement  2x3 
sq.  in.  =  6  sq.  in.  A  parallelogram  cut  from  paper 
is  easily  shown  by  the  use  of  the  scissors  to  equal 
in  area  the  rectangle  of  the  same  base  and  same 
altitude,  a  figure  already  considered.  By  paper-cut- 
ting the  triangle  is  shown  to  be  equal  to  half  of  a 
certain  parallelogram,  and  hence  to  half  of  the  rec- 
tangle having  the  same  base  and  the  same  altitude. 
By  a  few  measurements  of  circumferences  and  their 
corresponding  diameters  the  ratio  c  :  d  can  be  shown 
to  be  approximately  3-^,  a  value  sufficiently  exact  for 
ordinary  mensuration.  The  teacher  may  then  state, 
if  thought  best,  that  it  is  proved  in  geometry  that  a 
closer  approximation  is  3.1416,  or  3.14159.  The  pupil 
has  thus  the  interest  of  a  partial  discovery,  and  at 
the  same  time  the  possibilities  of  the  more  advanced 
mathematics  are  suggested.  Similarly,  as  set  forth  in 


THE  PRESENT  TEACHING  OF  ARITHMETIC         139 

many  of  our  better  class  of  text-books,  the  other 
necessary  propositions  in  mensuration  may  profitably 
be  treated.1 

Text-books  —  In  the  days  when  text-books  were  few 
and  poor  there  was  some  excuse  for  dictating  elab- 
orate notes.  The  arithmetic  copy-book  was  then  an 
institution  of  some  importance.  But  at  present  there 
is  no  such  excuse;  we  have  good  books,  and  they 
save  the  time  of  pupil  and  teacher.  This  does  not 
mean  that  the  book  shall  be  a  master  to  be  feared, 
but  rather  a  servant  to  assist.  In  the  lower  grades, 
while  the  teacher  should  seek  to  follow  the  general 
lines  of  the  text-book,  each  new  demonstration  should 
be  discovered  by  the  class  (of  course  with  the  teacher's 
leading)  in  advance  of  the  assignment  of  book  work. 
If  the  author's  plan  is  reasonably  satisfactory  it  should 
be  followed,  in  order  that  the  pupil  may  be  able  to 
review  the  discussion  without  the  waste  of  time  in 
note-taking;  a  great  many  hours  are  squandered  by 
teachers  in  attempting  to  "develop"  something  along 
some  line  not  followed  by  the  text-book  in  hand, 
when  the  author's  method  is  quite  as  good  —  usually 
better.  There  are  now  several  excellent  text-books 
with  satisfactory  demonstrations  and  with  up-to-date 
problems,  and  these  should  receive  the  support  of  the 
profession. 

1  See  also  Ilanus,  P.  II.,  Geometry  in  the  Grammar  School,  Boston, 
1893. 


140    THE  TEACHING  OF  ELEMENTARY   MATHEMATICS 

But  with  any  text-book  we  shall  do  well  to  keep 
in  mind  the  words  of  President  Hall :  "  American 
teachers  seem  to  me  to  have  spun  the  simple  and 
immediate  relations  and  properties  of  numbers  over 
with  pedantic  difficulties.  The  four  rules,  fractions, 
factoring,  decimals,  proportion,  per  cent.,  and  roots, 
is  not  this  all  that  is  essential?  The  best  European 
text-books  I  know  do  only  this,  and  are  in  the 
smaller  compass,  for  they  look  only  at  facility  in 
pure  number  relations,  which  is  hindered  by  the  irrele- 
vant material  which  publishers  and  bad  teachers  use 
as  padding."1 

Explanations  —  The  question  of  the  explanations  to 
be  given  to  and  demanded  from  a  child  is  a  serious 
one.  The  primary  work  is  preeminently  that  of  lead- 
ing the  child  to  discover  the  relations  of  number,  and 
to  memorize  certain  facts  (like  the  multiplication  table) 
which  he  will  subsequently  need.  A  few  rules  of 
action  suggested  by  M.  Laisant  are  worthy  of  atten- 
tion :  "  Follow  a  rigorously  experimental  method  and 
do  not  depart  from  it;  leave  the  child  in  the  pres- 
ence of  concrete  realities  which  he  sees  and  handles 
to  make  his  own  abstractions ;  never  attempt  to 
demonstrate  anything  to  him;2  merely  furnish  to  him 
such  explanations  as  he  is  himself  led  to  ask;  and 

1  Letter  from  G.  Stanley  Hall  to  F.  A.  Walker,  in  the  latter's  monograph 
on  arithmetic,  p.  23. 

a  /.£.,  by  a  formal,  logical  demonstration. 


THE  PRESENT  TEACHING  OF  ARITHMETIC         141 

finally,  give  and  preserve  to  this  teaching  an  appear- 
ance of  pleasure  rather  than  of  a  task  which  is  im- 
posed. If  cerebral  fatigue  is  produced,  if  the  child 
is  led  to  fix  his  attention  on  matters  of  no  interest, 
and  to  master  a  line  of  reasoning  too  much  in  ad- 
vance for  him,  then  the  result  is  a  failure."  1 

The  period  of  explanation  comes  later  in  the  course, 
say  after  the  fifth  grade;  but  even  here  the  explana- 
tion should  rather  be  by  questioning  on  the  part  of 
the  teacher  than  by  a  full  and  free  demonstration  by 
the  pupil.  Where  complete  "  explanations "  are  re- 
quired from  the  pupil,  say  of  subjects  like  greatest 
common  divisor,  the  division  of  fractions,  cube  root, 
etc.,  the  result  is  usually  a  lot  of  memoriter  work  of 
no  more  value  than  the  repetition  of  a  string  of  rules. 
But  by  questioning  as  to  the  "why"  of  the  various 
steps,  the  reasoning  (which  in  most  such  work  is  all 
that  is  essential)  is  laid  bare. 

It  is  the  same  with  many  applied  problems.  The 
set  forms  of  analysis  sometimes  required  of  pupils 
is  of  very  questionable  value.  On  the  other  hand,  a 
statement  of  the  pupil's  own  reasoning  is,  of  course, 
extremely  important,  when  he  is  sufficiently  advanced 
to  give  it.  But  for  primary  children  any  elaborate 
explanation  is  impossible.  Indeed,  in  the  midst  of  all 
our  theorizing  on  the  subject  of  explanations,  it  is 
refreshing  to  read  what  a  psychologist  like  Professor 

1  La  Math£matique,  p.  203,  204. 


142    THE  TEACHING  OF  ELEMENTARY  MATHEMATICS 

James  has  to  say  upon  the  subject  of  primary  work : 
"  It  is  ...  in  the  association  of  concretes  that  the 
child's  mind  takes  most  delight.  Working  out  results 
by  rule  of  thumb,  learning  to  name  things  when  they 
see  them,  drawing  maps,  learning  languages,  seem  to 
me  the  most  appropriate  activities  for  children  under 
thirteen  to  be  engaged  in.  ...  I  feel  pretty  confi- 
dent that  no  man  will  be  the  worse  analyst  or  reasoner 
or  mathematician  at  twenty  for  lying  fallow  in  these 
respects  during  his  entire  childhood."1 

Approximations  —  There  is  a  feeling  among  many 
teachers  that  some  virtue  attaches  to  the  carrying  of 
a  result  to  a  large  number  of  decimal  places,  and 
hence  this  is  rather  encouraged  among  pupils.  As 
a  matter  of  fact  the  contrary  is  usually  the  case  in 
practice.  If  the  diameter  of  a  circle  has  been  meas- 
ured correctly  to  o.ooi  inch  there  is  no  use  in 
attempting  to  compute  the  circumference  to  more 
than  three  decimal  places,  and  3.1416  is  a  better  mul- 
tiplier than  3.14159.  The  result  should  be  cut  off 
at  thousandths  and  the  labor  of  extending  it  beyond 
that  place  should  be  saved. 

Now  since  we  rarely  use  decimals  beyond  o.ooi 
except  in  scientific  work,  and  since  no  result  can  be 
more  exact  than  the  data,  and  since  even  our  scientific 
measurements  rarely  give  us  data  beyond  three  or  four 
decimal  places,  the  practical  operations  are  the  contracted 

1  Letter  to  F.  A.  Walker,  in  the  latter's  monograph,  p.  22. 


THE  PRESENT  TEACHING  OF  ARITHMETIC          143 

ones,  those  which  are  correct  to  a  given  number  of 
places.  For  this  reason,  in  this  age  of  science,  ap- 
proximate methods  are  of  great  value  in  the  higher 
grades  which  precede  the  study  of  physics.  The  fol- 
lowing are  types  of  such  work:1 

10.48 
3.1416)32.92  =     31416)329200 


150 
126 

24 
24 

32.92 

For  the  same  reason  the  practical  use  of  a  small 
logarithmic  table  is  of  great  value  in  the  computa- 
tions of  elementary  physics.  Two  or  three  lessons 
suffice  to  explain  the  use  of  the  tables  and  to  justify 
the  laws  of  operation,  a  small  working  table  can  be 
bought  for  five  cents,  and  the  field  of  physics  affords 
abundant  practice. 

Reviews  —  However  much  reviews  may  fail  from 
their  stupidity,  as  is  apt  to  be  the  case  with  "set 
reviews,"  a  skilful  teacher  is  always  reviewing  in 
connection  with  the  advance  work.  But  there  is  one 
season  when  a  review  is  essential,  a  brisk  running 

1  The  explanations  are  given  in  any  higher  arithmetic,  e.g.  Beman 
and  Smith,  p.  8,  u. 


144    THE  TEACHING  OF  ELEMENTARY   MATHEMATICS 

over  of  the  preceding  work  that  the  pupil  may  take 
his  bearings,  and  this  is  at  the  opening  of  the  school 
year.  Such  a  refreshening  of  the  mind,  such  a  lubri- 
cating of  the  mental  machinery,  gets  one  ready  for 
the  year's  work.  Complaints  which  teachers  generally 
make  of  poor  work  in  the  preceding  grade  are  not 
unfrequently  due  to  the  one  complaining;  the  effects 
of  the  long  vacation  have  been  forgotten;  the  engine 
is  rusty  and  it  needs  oiling  before  the  serious  start 
is  made. 

In  these  reviews  the  same  correctness  of  statement 
is  necessary  as  in  the  original  presentation,  though 
not  always  the  same  completeness.  To  let  a  child 
say  that  2  +  3x2  is  10  (instead  of  8)  is  to  sow  tares 
which  will  grow  up  and  choke  the  good  wheat.  To 
let  him  see  forms  like 

2  ft.  x  3  ft.  =  6  sq.  ft.,  45°  -j-  15  =  3  hrs., 
V4  sq.  ft.  =  2  ft,  2  x  0.50  =  $  i,  etc., 

or  to  let  him  hear  expressions  like  "As  many  times 
as  2  is  contained  in  $  10,"  "  2  times  greater  than  $  3," 
etc.,  is  to  take  away  a  large  part  of  the  value  that 
mathematics  should  possess. 


CHAPTER  VI 
THE  GROWTH  OF  ALGEBRA 

Egyptian  algebra  —  Reserving  for  the  following 
chapter  the  question  of  the  definition  of  algebra,  we 
may  say  that  the  science  is  by  no  means  a  new  one. 
Or  rather,  to  be  more  precise,  the  idea  of  the  equa- 
tion is  not  new,  for  this  is  only  a  part  of  the  rather 
undefined  discipline  which  we  call  algebra.  In  the 
oldest  of  extant  deciphered  mathematical  manuscripts, 
the  Ahmes  papyrus  to  which  reference  has  already 
been  made,  the  simple  equation  appears.  It  is  true 
that  neither  symbols  nor  terms  familiar  in  our  day 
are  used,  but  in  the  so-called  hau  computation  the 
linear  equation  with  one  unknown  quantity  is  solved. 
Symbols  for  addition,  subtraction,  equality,  and  the 
unknown  quantity  are  used.  The  following  is  an 
example  of  the  simpler  problems  which  Ahmes  gives, 
his  twenty-fourth :  "  Hau  (literally  heap\  its  seventh, 
its  whole,  it  makes  19,"  which  put  in  modern  sym- 
bols means  -  +  x  =  19.  Somewhat  more  difficult 

problems    are    also    given,    like    the    following    (his 
thirty-first):    "Hau,   its   f,   its   J,   its   \t   its   whole,   it 

makes  33," 

i.e.,  %x  +  \x  + 

L  145 


146     THE  TEACHING  OF  ELEMENTARY   MATHEMATICS 

It  must  be  said,  however,  that  Ahmes  had  no 
notion  of  solving  the  equation  by  any  of  our  present 
algebraic  methods.  His  was  rather  a  "  rule  of  false 
position,"  as  it  was  called  in  mediaeval  times,  —  guess- 
ing at  an  answer,  rinding  the  error,  and  then  modi- 
fying the  guess  accordingly.1  Ahmes  also  gives 
some  work  in  arithmetical  series  and  one  example  in 
geometric. 

Greek  algebra  —  Algebra  made  no  further  progress, 
so  far  as  now  known,  among  the  Egyptians.  But  in 
the  declining  generations  of  Greece,  long  after  the 
"golden  age"  had  passed,  it  assumed  some  impor- 
tance. As  already  stated,  the  Greek  mind  had  a 
leaning  toward  form,  and  so  it  worked  out  a  wonder- 
ful system  of  geometry  and  warped  its  other  mathe- 
matics accordingly.  The  fact  that  the  sum  of  the 
first  n  odd  numbers  is  «2,  for  example,  was  dis- 
covered or  proved  by  a  geometric  figure ;  square  root 
was  extracted  with  reference  to  a  geometric  diagram; 
figurate  numbers  tell  by  their  name  that  geometry 
entered  into  their  study. 

So  we  find  in  Euclid's  "  Elements  of  Geometry " 
(B.C.,  c.  300)  formulae  for  (a  +  ftf  and  other  simple 
algebraic  relations  worked  out  and  proved  by  geo- 
metric figures.  Hence  Euclid  and  his  followers  knew 

1  Besides  Eisenlohr's  translation  already  mentioned,  see  Cantor,  I, 
p.  38.  A  short  sketch  is  given  in  Gow's  History  of  Greek  Mathematics, 
p.  18. 


THE  GROWTH  OF  ALGEBRA  147 

from  the  figure  that  to  "  complete  the  square,"  the 
geometric  square,  of  x*  +  2  ax,  it  is  necessary  to  add 
a2.  He  also  solved,  geometrically,  quadratic  equations 
of  the  form  ax  —  x*=b>  ax  +  x*  =  b,  and  simultaneous 
equations  of  the  form  x  ±  y  =  a,  xy=bl 

With  the  older  Greek  view  of  mathematics, 
ever,  it  was  impossible  for  algebra  to  make  much 
headway.  Recognizing  the  linear,  quadratic,  and 
cubic  functions  of  a  variable,  because  these  could  be 
represented  by  lines,  squares,  and  cubes,  the  Greeks 
of  Euclid's  time  refused  to  consider  the  fourth  power 
of  a  variable  because  the  fourth  dimension  was 
beyond  their  empirical  space. 

Algebra  had,  however,  made  a  beginning  before 
Euclid's  time.  Thymaridas  of  Paros,  whose  personal 
history  is  quite  unknown,  had  already  solved  some- 
simple  equations,  and  had  been  the  first  to  use  the 
expressions  given  or  defined  (apurpevoi),  and  unknown^ 
or  undefined  (ao/ato-rot),2  and  it  seems  not  improbable 
that  the  quadratic  equation  was  somewhat  familiar 
before  the  Alexandrian  school  was  founded.8  Aris- 
totle, too,  had  employed  letters  to  indicate  unknown 
quantities  in  the  statement  of  a  problem,  although 
not  in  an  equation.4 

1  Heath,  T.  L.,  Diophantos  of  Alexandria,  Cambridge,  1885,  p.  140. 

1  Cantor,  I,  p.  148  ;  Gow,  p.  97,  107. 

»  Cantor,  I,  p.  301  ;  but  see  Heath's  Diophantos,  p.  139. 

*  Gow,  p.  105. 


148    THE  TEACHING  OF  ELEMENTARY  MATHEMATICS 

The  most  notable  advance  before  the  Christian  era 
was  made  by  Heron  of  Alexandria,  about  100  B.C. 
Breaking  away  from  the  pure  geometry  of  his  prede- 
cessors, and  not  hesitating  to  speak  of  the  fourth 
power  of  lines,  he  solved  the  quadratic  equation1  and 
ran  up  against  imaginary  roots.2  This  was  the 
turning-point  of  Greek  mathematics,  the  downfall  of 
their  pure  geometry,  the  rise  of  a  new  discipline. 

But  it  is  to  Diophantus  that  we  owe  the  first 
serious  attempt  to  work  out  this  new  science.  An 
Alexandrian,  living  in  the  fourth  century,  probably  in 
the  first  half,  he  wrote  a  work,  'A/>i#/-t?7Ti/ea,  almost 
entirely  devoted  to  algebra.3  This  work  is  the  first 
one  known  to  have  been  written  upon  algebra  alone 
(or  chiefly).  Diophantus  uses  only  one  unknown 
quantity,  6  apiOpos  or  o  adptoTo?  apt^/xo?,  symbolizing 
it  by  9'  or  5°'.4  The  square  he  calls  Swapis,  power 
(its  symbol  8°),  the  cube  /cvpos  (/c°),  and  he  also  gives 
names  to  the  fourth,  fifth,  and  sixth  powers.  He 
has  symbols  for  equality  and  for  subtraction,  and  the 
modern  expression  x*  —  $x2  +  8x  —  i  he  would  write 

1  Cantor,  I,  p.  377  ;  Gow,  p.  106. 

2  Cantor,  I,  p.  374  ;  Beman,  W.  W.,  vice-presidential  address,  Section 
A,  American  Assoc.  Adv.  Sci.,  1897. 

3  Heath,  T.  L.,  Diophantos   of  Alexandria,  Cambridge,  1885  ;    Gow, 
p.  100 ;   Hankel  and  Cantor,  of  course,  on  all  such  names.     De  Morgan 
has  a  good  article  on  Diophantus  in  Smith's  Diet,  of  Gk.  and  Rom.  Biog., 
a  work  containing  several  valuable  biographies  of  mathematicians. 

*  For  discussion  of  the  symbol,  see  Heath,  p.  56-66. 


THE  GROWTH  OF  ALGEBRA  149 

in  the  form  /c"d<;olrjsjiS"efji°dtl  a  form  not  particularly 
more  difficult  than  our  own.  The  nature  of  his  solu- 
tions will  be  understood  from  the  following  example, 
modern  symbols  being  here  used :  "  Find  two  num- 
bers whose  sum  is  20  and  the  difference  of  whose 
squares  is  80.  • 

Put  for  the  numbers  x+  10,  10  —  x. 

Squaring,  we  have  x*  +  20  x  +  100,  x*  +  100  —  20  x. 

The  difference,         40  x  =  So. 

Dividing,  x  =  2. 

Result,  greater  is  12,  less  is  8."2  This  does  not 
differ  from  our  own  present  plan,  although  being  less 
troubled  by  negative  numbers  we  would  probably  say : 

(20  -  x?  -  x*  =  80. 
/.  400-40*  =  80. 

.-.  320  =  40*. 
/.  8  =  *,  and  20  —  *=  12. 

It  thus  appears  that  Diophantus  understood  the 
simple  equation  fairly  well.  The  quadratic,  however,  he 
solved  merely  by  rule.  Thus  he  says,  "84^—7^=7, 
therefore  x  =  £,"  giving  but  one  of  the  two  roots. 
Of  the  negative  quantity  he  apparently  knew  nothing, 
and  his  work  was  limited,  with  the  exception  of  a 
single  easy  cubic,  to  equations  of  the  first  two 
degrees.  His  favorite  subject  was  indeterminate 

1  Heath,  p.  72.  2  Ib.,  p.  76. 


150    THE  TEACHING  OF  ELEMENTARY   MATHEMATICS 

equations  of  the  second  degree,  and  on  this  account 
indeterminate  equations  in  general  are  often  desig- 
nated as  Diophantine.  One  of  the  most  remarkable 
facts  connected  with  the  work  of  Diophantus  is  that, 
although  most  other  algebraists  down  to  about  1700 
•LD.,  used  geometric  figures  more  or  less,  he  nowhere 
appeals  to  them.1  Summing  up  the  work  of  the 
Greeks  in  this  field,  we  may  say  that  they  could* 
solve  simple  and  quadratic  equations,  could  represent 
geometrically  the  positive  roots  of  the  latter,  and 
could  handle  indeterminate  equations  of  the  first  and1 
second  degrees. 

Oriental  algebra  —  It  was  long  after  the  time  of 
Diophantus,  and  in  a  country  well  removed  from 
Greece,  and  among  a  race  greatly  differing  from  the 
Hellenic  people,  that  algebra  took  its  next  noteworthy 
step  forward.  It  is  true  that  Aryabhatta,  a  Hindu 
mathematician  (b.  476),  made  some  contributions  to  the 
subject  not  long  after  Diophantus  wrote,  but  he  did  not 
carry  the  subject  materially  farther  than  the  Greeks,2 
and  it  was  not  until  about  800  A.D.  that  the  next  real 
advance  was  made. 

When  under  the  Calif  Al-Mansur  (the  Victorious, 
c.  712 -775)  it  was  decided  to  build  a  new  capital  for 

1  Gow,  p.  114  n. ;   Hankel,  p.  162. 

2  Cantor,  I,  p.  575;   Hankel,  p.  172;  Matthiessen,  L.,  Grundziige  der 
antiken  und  modernen  Algebra  der  litteralen  Gleichungen,  2.  Ausg.,  Leip- 
zig, 1896,  p.  967. 


THE  GROWTH  OF  ALGEBRA  151 

the  Mohammedan  rulers,  the  site  of  an  ancient  city 
dating  back  to  Nebuchadnezzar's  time,  on  the  banks  of 
the  Tigris,  was  chosen.  To  this  new  city  of  Bagdad 
were  called  scholars  from  all  over  the  civilized  world, 
Christians  from  the  West,  Buddhists  from  the  East, 
and  such  Mohammedans  as  might,  in  those  early  days, 
of  that  religion,  be  available.  With  this  enlightened 
educational  policy,  a  policy  opposed  to  in-breeding  and 
to  sectarianism,  Bagdad  soon  grew  to  be  the  centre  of 
the  civilization  of  that  period.  Under  Harun-al-Raschid 
(Aaron  the  Just,  calif  from  786  to  809)  the  calif  ate 
reached  the  summit  of  its  power,  extending  from  the 
Indus  to  the  Pillars  of  Hercules.  His  son  Al-Mamun 
(786-833),  whom  Sismondi  calls  "the  father  of  letters 
and  the  Augustus  of  Bagdad,"  brought  Arab  learning 
to  its  height.  It  was  during  his  reign,  in  the  first  quarter 
of  the  ninth  century,  that  there  came  from  Kharezm 
(Khwarazm),  a  province  of  Central  Asia,  a  mathemati- 
cian known  from  his  birthplace  as  Al-Khowarazmi.1 
He  wrote  the  first  general  work  of  any  importance  on 
algebra,  that  of  Diophantus  being  largely  confined  to  a 
single  class  of  equations,  and  to  the  science  he  gave  its 
present  name.  He  designated  it  Ilm  al-jabr  wa'l  mu- 
qabalak,  that  is,  "the  science  of  redintegration  and 
equation,"  a  title  which  appeared  in  the  thirteenth  cen- 
tury Latin  as  Indus  algebra  almucgrabalceque,  in  six- 

JAbu  Ja'far  Mohammed  ben  Musa  al-Khowarazmi,  Abu  Ja'far  Moham- 
med son  of  Moses  from  Kharezm.    Cantor,  I,  p.  670. 


152    THE  TEACHING  OF  ELEMENTARY   MATHEMATICS 

teenth  century  English  as  algiebar  and  almachabel,  and 
in  modern  English  as  algebra?  So  important  were  also 
his  writings  on  arithmetic,  that  just  as  "  Euclid "  is  in 
England  a  synonym  for  elementary  geometry,  so  algo- 
ritmi  (from  al-Khowarazmi)  was  for  a  long  time  a  syn- 
onym for  the  science  of  numbers,  a  word  which  has 
survived  in  our  algorism  (algorithm). 

Al-Khowarazmi  discussed  the  solution  of  simple 
and  quadratic  equations  in  a  scientific  manner,  dis- 
tinguishing six  different  classes,  much  as  our  old-style 
writers  on  arithmetic  distinguished  the  various  "cases" 
of  percentage.  His  classes  were,  in  modern  notation, 
ax*  =  bx,  ax*  =  c,  bx  =  c,  x*  4-  bx  —  c,  x 2  +c  =  bxy  x*= 
bx  -f  c, 2  showing  how  primitive  was  the  science  which 
could  not  grasp  the  general  type  ax 2  -f-  bx  +  c  —  o. 
His  method  of  stating  and  solving  a  problem  may 
be  seen  in  the  following : 3  "  Roots  and  squares  are 
equal  to  numbers ;  for  instance,  one  square  and  ten 
roots  of  the  same  amount  to  thirty-nine ; 4  that  is  to 
say,  what  must  be  the  square  which,  when  increased 
by  ten  of  its  own  roots,  amounts  to  thirty-nine  ?  The 
solution  is  this :  you  halve  the  number  of  the  roots, 
which  in  the  present  instance  yields  five.  This  you 
multiply  by  itself;  the  product  is  twenty-five.  Add 

1  See  also  Heath,  p.  149.  2  Cantor,  I,  p.  676. 

8  From  The  Algebra  of  Mohammed-ben-Musa,  edited  and  translated 
by  Frederic  Rosen,  London,  1831. 
10*  =  39. 


THE  GROWTH  OF  ALGEBRA  153 

this  to  thirty-nine  ;  the  sum  is  sixty-four.  Now  take 
the  root1  of  this,  which  is  eight,  and  subtract  from 
it  half  the  number  of  the  root,  which  is  five;  the 
remainder  is  three.  This  is  the  root  of  the  square 
for  which  you  sought."2  The  solution  merely  sets 
forth  without  explanation  the  rule  expressed  in  our 
familiar  formula  for  the  solution  of  x*  +px  +  q  =  o, 

i.e.,  x  =  —  —  -±  J  V/2—  4^,   except   that   only  one  root 

is  given.  He  however  recognizes  the  existence  of 
two  roots  where  both  are  real  and  positive,  as  in  the 
equation  x*  +  21  =  io;r.3  In  practice  he  commonly 
uses  but  one  root. 

Sixteenth  century  algebra  —  Algebra  made  little  ad- 
vance, save  in  the  way  of  the  solution  of  a  few  special 
cubics,  from  the  time  of  Mohammed  ben  Musa  to  the 
sixteenth  century,  seven  hundred  years.  Its  course 
had  run  from  Egypt  to  Greece,  and  from  Greece  (and 
Grecian  Alexandria)  to  Persia.  It  now  transfers  itself 
from  Persia  to  Italy  and  works  slowly  northward. 

In  a  famous  work  printed  in  Nurnberg  in  1545, 
the  "Ars  magna,"4  Cardan  gives  a  complete  solution  of 
a  cubic  equation  ;  that  is,  he  solves  an  equation  of  the 

«. 

1  />.,  the  square  root 

2  The    successive    steps    are   as    follows:  \  of  10  =  5;  5.5  =  25;  25 
+  39  =  64;   V^4  =  8;  8-5  =  3. 

8  Rosen,  p.   1  1. 

4  Hieronymi  Cardani,  praestantissimi  mathematici,  philosophi,  ac 
medici,  Artis  Magnoe,  sive  de  regvlis  algcbraicis,  Lib.  unus. 


OF  THE 

UNIVERSITY 


154    THE  TEACHING  OF   ELEMENTARY  MATHEMATICS 

form  x?+px=q,  to  which  all  other  cubics  can  be 
reduced.  He  mentions,  however,  his  indebtedness  to 
earlier  writers,  though  not  as  generously  as  seems  to 
have  been  their  due.1 

This  is  not  the  place  to  consider  the  relative  claims 
of  Cardan,  Tartaglia  (Tartalea),  Ferro  (Ferreus), 
and  Fiori  (Florido).  Cardan  seems  to  have  obtained 
Tartaglia's  solution  of  the  cubic  under  pledge  of 
secrecy  and  then  to  have  published  it.  But  however 
this  was,  by  the  middle  of  the  sixteenth  century  the 
cubic  equation  was  solved,  and  Ludovico  Ferrari  at 
about  the  same  time  solved  the  quartic. 

Algebra  had  now  reached  such  a  point  that  mathema- 
ticians were  able  to  solve,  in  one  way  or  another,  gene- 
ral equations  of  the  first  four  degrees.  Thereafter  the 
chief  improvements  were  (i)  in  symbolism,  (2)  in  under- 
standing the  number  system  of  algebra,  (3)  in  finding 
approximate  roots  of  higher  numerical  equations,  (4)  in 
simplifying  the  methods  of  attacking  equations,  and  (5) 
in  the  study  of  algebraic  forms.  For  the  purposes  of 
elementary  algebra  we  need  at  this  time  to  speak  only 
of  the  first  three. 

1  Scipio  Ferreus  Bononiensis  iam  annis  ab  hinc  triginta  ferine  capit- 
ulum  hoc  inuenit,  tradidit  uero  Anthonio  Mariae  Florido  Veneto,  qui 
cu  in  certamen  cu  Nicolao  Tartalea  Brixellense  aliquando  uenisset, 
occasionem  dedit,  ut  Nicolaus  inuenerit,  &  ipse,  qui  cum  nobis  rogan- 
tibus  tradidisset,  supprcssa  demonstrationc,  freti  hoc  auxilio,  demonstra- 
tionem  qusesiuimus,  eamque  in  modos,  quod  difficillimum  fuit,  redactam 
sic  subiecimus.  Fol.  29,  v. 


THE  GROWTH  OF  ALGEBRA  155 

Growth  of  symbolism  —  Algebra,  as  is  readily  seen,  is 
very  dependent  upon  its  symbolism.  Its  history  has 
been  divided  into  three  periods,  of  rhetorical,  of  synco- 
pated, and  of  symbolic  algebra.  The  rhetorical  algebra 
is  that  in  which  the  equation  is  written  out  in  words,  as 
in  the  example  given  on  p.  152  from  Al-Khowarazmi ; 
the  syncopated,  that  in  which  the  words  are  abbre- 
viated, as  in  most  of  the  example  given  on  p.  149 
from  Diophantus;  the  symbolic,  that  in  which  an 
arbitrary  shorthand  is  used,  as  in  our  common  algebra 
of  to-day. 

The  growth  of  symbolism  has  been  slow.  From  the 
radical  sign  of  Chuquet  (1484),  R4.  10,  through  various 
other  forms,  as  V^  IO>  to  our  common  symbol,  v  10 
and  to  the  more  refined  10*,  which  is  only  slowly  becom- 
ing appreciated  in  elementary  schools,  is  a  tedious  and  a 
wandering  path.  So  from  Cardan's 

cubus  p  6.  rebus  aequalis  20,  for  ;r3+6;r=2O, 
through  Vieta's 

iC  -  8  Q  +  16  N  aequ.  40,  for  x3  —  Sx?  +  16^=40, 
and  Descartes's 

x*  x  ax—  bb,  for  x* — ax—  IP, 
and  Hudde's 

x*  so  qx.r,  for  x^—  qx-\-r^ 

1  Beman  and  Smith's  translation  of  Fink's  History  of  Mathematics, 
p.  108. 


156    THE  TEACHING  OF  ELEMENTARY  MATHEMATICS 

has  likewise  been  a  long  and  tiresome  journey.  Such 
simple  symbols  as  the  x  for  multiplication,1  and  the  still 
simpler  dot  used  by  Descartes,  the  =  for  equality,2 

the  x~*  for  — ,3  these  all  had  a  long  struggle  for  recog- 
nition. Even  now  the  symbol  -*-  has  only  a  limited 
acceptance  in  the  mathematical  world,  and  there  are 
three  widely  used  forms  for  the  decimal  point.4  Thus 
symbolism  has  been  a  subject  of  slow  growth,  and  we 
are  still  in  the  period  of  unrest. 

We  may,  however,  assign  to  the  Frenchman  Vieta5 
the  honor  of  being  the  founder  of  symbolic  algebra  in 
large  measure  as  we  recognize  it  to-day.  His  first  book 
on  algebra,  "  In  artem  analyticam  isagoge,"  appeared  in 
i59i.6  Laisant  thus  summarizes  his  contribution  :  "  He 
it  is  who  should  be  looked  upon  as  the  founder  of  alge- 
bra as  we  conceive  it  to-day.  The  powerful  impulse 
which  he  gave  consisted  in  this,  that  while  unknown 
quantities  had  already  been  represented  by  letters  to 
facilitate  writing,  it  was  he  who  applied  the  same  method 
to  known  quantities  as  well.  From  that  day,  when  the 
search  for  values  gave  way  to  the  search  for  the  opera- 
tions to  be  performed,  the  idea  of  the  mathematical 

1  First  used  by  Oughtred  in  1631. 

2  Recorde,  1556.  8Wallis. 

4  2-|  is  usually  written  2.5  in  America,  2.5  in  England,  2,5  on  the  Con- 
tinent. 

5  Francois  Viete,  1540-1603. 

6  Cantor,  II,  p.  577;  for  a  general  summary  of  his  work,  see  p.  595. 


THE  GROWTH  OF  ALGEBRA  157 

function  enters  into  the  science,  and  this  is  the  source  of 
its  subsequent  progress."  l 

Number  systems  —  The  difficulty  of  understanding 
the  number  systems  of  algebra  has  been,  perhaps,  the 
greatest  obstacle  to  its  progress.  The  primitive, 
natural  number  is  the  positive  integer.  So  long  as 
the  world  met  only  problems  which  may  be  repre- 
sented by  the  modern  form  ax  -f  b  =  c,  where  c  >  b 
and  c  —  b  is  a  multiple  of  a,  as  in  3  x  +  2  =  1 1,  these 
numbers  sufficed.  But  when  problems  appeared  which 
involve  the  form  of  equation  ax—b  where  b  is  not 
a  multiple  of  a,  as  in  3^r=  i,  or  2,  or  5,  then  other 
kinds  of  number  are  necessary,  the  unit  fraction,  the 
general  proper  fraction,  and  the  improper  fraction  or 
mixed  number.  We  have  seen  (Chap.  Ill)  how  the 
world  had  to  struggle  for  many  centuries  before  it  came 
to  understand  numbers  of  this  kind.  It  was  only  by 
an  appeal  to  graphic  methods  (the  representation  of 
numbers  by  lines)  that  the  fraction  came  to  be  under- 
stood. When,  further,  problems  requiring  the  solution 
of  an  equation  like  x*=a,a  not  being  an  nth  power,  as 
in  x*  =  2,  still  a  new  kind  of  number  was  necessary,  the 
real  and  irrational  number,  a  form  which  the  Greeks 
interpreted  geometrically  for  square  and  cube  roots. 

The  next  step  led  to  equations  like  x  4-  a  =  b,  with 
a  >  £,  as  in  x  +  5  =  2,  a  form  which  for  many  centuries 
baffled  mathematicians  because  they  could  not  bring 

1  La  Math6matique,  p.  55. 


158    THE  TEACHING  OF  ELEMENTARY  MATHEMATICS 

themselves  to  take  the  step  into  the  domain  of  nega- 
tive numbers.  It  was  not  until  the  genius  of  Des- 
cartes (1637)  more  completely  grasped  the  idea  of 
the  one-to-one  correspondence  between  algebra  and 
geometry,  that  the  negative  number  was  taken  out 
of  the  domain  of  numerce  jictcz1  and  made  entirely 
real.  One  more  step  was,  however,  necessary  for 
the  solution  of  equations  of  the  form  xn  +  a  =  o. 
What  to  do  with  an  equation  like  ^  +  4  =  0  was  still 
an  unanswered  question.  To  say  that  x  —  V—  4,  or 
2V  —  i,  or  ±  2V—  I,  avails  nothing  unless  we  know 
the  meaning  of  the  symbol  "V— i."  It  was  not 
until  the  close  of  the  eighteenth  century  that  any 
considerable  progress  was  made  in  the  interpretation 
of  the  symbol  a  -f  £V—  i.  In  1797  Caspar  Wessel, 
a  Norwegian,  suggested  the  modern  interpretation,  and 
published  a  memoir  upon  complex  numbers  in  the 
proceedings  of  the  Royal  Academy  of  Sciences  and 
Letters  of  Denmark  for  I797-2  Not,  however,  until 
Gauss  published  his  great  memoir  on  the  subject 
(1832)  was  the  theory  of  the  graphic  representation  of 

1  Cardan,  Ars  magna,  1545,  Fol.  3,  v. 

2  This  has  recently  been  republished  in  French  translation,  under  the 
title  Essai  sur  la  representation  analytique  de  la  direction,  Copenhague, 
1897,  witn  a  historical  preface  by  H.  Valentiner.     For  a  valuable  summary 
of  the  history,  see  the  vice-presidential  address  of  Professor  Beman,  Section 
A  of  the  American  Assoc.  Adv.  Sci.,  1897.     A  brief  summary  is  also 
given  in  the  author's  History  of  Modern  Mathematics,  in  Merriman  and 
Woodward's  Higher  Mathematics,  New  York,  1896. 


THE  GROWTH  OF  ALGEBRA  159 

the  complex  number  generally  known  to  the  mathemat- 
ical world.  Elementary  text-book  writers  still  seem 
indisposed  to  give  the  subject  place,  although  its 
presentation  is  as  simple  as  that  of  negative  numbers.1 
For  the  purposes  of  elementary  teaching  only  a 
single  other  historical  question  demands  consideration, 
the  approximate  solution  of  numerical  equations,  and 
even  this  is  rather  one  of  arithmetic  than  of  algebra. 
Algebra  has  proved  that  there  is  no  way  of  solving 
the  general  equation  of  degree  higher  than  four  ;  that 
is,  that  by  the  common  operations  of  algebra  we  can 
solve  the  equation 

ax*  +  bx*  +  ex*  -f  dx  +  e  =  o, 
but  that  we  cannot  solve  the  equation 

axb  +  bx*  +  ex*  +  dx*  +  ex+f  =  o.2 
We  can,  however,  approximate  the  real  roots  of  any 
numerical    algebraic    equation,   and    this    suffices    for 
practical  work.     That  is,  we  can  find  that  one  root  of 
the  equation 

xb  -f  \2x*  +  59*3  +  i5O;r2  +  2iox—  207  =  o 
is  0.638605803+, 

but  we  have  no  formula  for  solving  such  equations  by 
algebraic  operations  as  we  have  for  solving 


1  For  an  elementary  treatment,  see  Beman  and  Smith's  Algebra,  Boston, 
1900. 

2  For  historical  resume,  see  the  author's  History  of  Modern  Mathematics 
already  cited,  p.  519. 


160    THE  TEACHING  OF  ELEMENTARY  MATHEMATICS 

The  simple  method  now  generally  used  for  this  ap- 
proximation is  due  to  an  Englishman,  W.  G.  Horner, 
who  published  it  in  1819,  and  it  now  appears  in  ele- 
mentary works  in  English  as  "  Horner's  method." 
Foreign  writers  have,  however,  been  singularly  slow 
in  recognizing  its  value. 


CHAPTER  VII 
ALGEBRA,  —  WHAT  AND  WHY  TAUGHT 

Algebra  defined  —  In  Chapter  VI  the  growth  of 
algebra  was  considered  in  a  general  way,  assuming 
that  its  nature  was  fairly  well  known.  Nor  is  it 
without  good  reason  that  this  order  was  taken,  for 
the  definition  of  the  subject  is  best  understood  when 
considered  historically.  But  before  proceeding  to  dis- 
cuss the  teaching  of  the  subject  it  is  necessary  to 
examine  more  carefully  into  its  nature. 

It  is  manifestly  impossible  to  draw  a  definite  line  be- 
tween the  various  related  sciences,  as  between  botany 
and  zoology,  between  physics  and  astronomy,  between 
algebra  and  arithmetic,  and  so  on.  The  child  who 
meets  the  expression  2  x  ( ? )  =  8,  in  the  first  grade, 
has  touched  the  elements  of  algebra.  The  student  of 
algebra  who  is  called  upon  to  simplify 

(2  +  V~3)/(2-V~3) 

is  facing  merely  a  problem  of  arithmetic.  In  fact, 
a  considerable  number  of  topics  which  are  prop- 
erly parts  of  algebra,  as  the  treatment  of  propor- 
tion, found  lodgment  in  arithmetic  before  its  sister 
science  became  generally  known;  while  much  of 
arithmetic,  like  the  theory  of  irrational  (including 
M  161 


1 62    THE  TEACHING  OF   ELEMENTARY   MATHEMATICS 

complex)  numbers,  has  found  place  in  algebra  simply 
because  it  was  not  much  needed  in  practical  arith- 
metic.1 

Recognizing  this  laxness  of  distinction  between  the 
two  sciences,  Comte2  proposed  to  define  algebra  "as 
having  for  its  object  the  resolution  of  equations ; 
taking  this  expression  in  its  full  logical  meaning, 
which  signifies  the  transformation  of  implicit  func- 
tions into  equivalent  explicit  ones.3  In  the  same  way- 
arithmetic  may  be  defined  as  destined  to  the  deter- 
mination of  the  values  of  functions.  Henceforth, 
therefore,  we  will  briefly  say  that  Algebra  is  the 
\  Calculus  of  Functions,  and  Arithmetic  the  Calculus  of 
\  Values."  4 

Of  course  this  must  not  be  taken  as  a  definition 
universally  accepted.  As  a  prominent  writer  upon 
"methodology"  says:  "It  is  very  difficult  to  give  a 

1  Teachers  who  care  to  examine  one  of  the  best  elementary  works  upon 
arithmetic  in  the  strict  sense  of  the  term,  should  read  Tannery,  Jules, 
Lemons  d'Arithmetique  theorique  et  pratique,  Paris,  1894. 

2  The  Philosophy  of  Mathematics,  translated  from  the  Cours  de  Philo- 
sophic positive,  by  W.  M.  Gillespie,  New  York,  1851,  p.  55. 

8  /.<?.,  in  x2  -f  px  +  q  =  o  we  have  an  implicit  function  of  x  equated  to 
zero  ;  this  equation  may  be  so  transformed  as  to  give  the  explicit  function 


and  this  transformation  belongs  to  the  domain  of  algebra. 

4  Laisant  begins  his  chapter  L'Algebre  (La  Mathematique,  p.  46)  by 
reference  to  this  definition,  and  makes  it  the  foundation  of  his  discussion 
of  the  science. 


ALGEBRA,  — WHAT  AND   WHY  TAUGHT  163 

good  definition  of  algebra.  We  say  that  it  is  merely 
a  generalized  or  universal  arithmetic,  or  rather  that 
'it  is  the  science  of  calculating  magnitudes  con- 
sidered generally'  (D'Alembert).  But  as  Poinsot  has 
well  observed,  this  is  to  consider  it  under  a  point  of 
view  altogether  too  limited,  for  algebra  has  two 
distinct  parts.  The  first  part  may  be  called  universal 
arithmetic.  .  .  .  The  other  part  rests  on  the  theory 
of  combinations  and  arrangement.  .  .  .  We  may 
give  the  following  definition.  .  .  .  Algebra  has  for  its 
object  the  generalizing  of  the  solutions  of  problems 
relating  to  the  computation  of  magnitudes,  and  of 
studying  the  composition  "siftd  transformations  of  for- 
mulae to  which  this  generalization  leads."  1  The  best 
of  recent  English  and  French  elementary  algebras 
make  no  attempt  at  defining  the  subject.2 

The  function  —  Taking  Comte's  definition  as  a  point 
of  departure,  it  is  evident  thatypne  of  the  first  steps 
in  the  scientific  teaching  of  algebra  is  the  fixing  of 
the  idea  of  futtction^.  How  necessary  this  is,  apart 
from  all  question  of  definition,  is  realized  by  all 
advanced  teachers.  "  I  found,"  says  Professor  Chrys- 
tal,  "when  I  first  tried  to  teach  university  students 
coordinate  geometry,  that  I  had  to  go  back  and 

1  Dauge,  Felix,  Cours  de  Methodologie  Mathematique,  2.  ed.,  Gand  et 
Paris,  1896,  p.  103. 

2  Chrystal,  G.,  2  vols.  2  ed.,  Edinburgh,  1889.      Bourlet,  C.,  Lemons 
d'Algebre  elementaire,  Paris,  1896. 


164    THE  TEACHING  OF  ELEMENTARY   MATHEMATICS 

teach  them  algebra  over  again.  The  fundamental 
idea  of  an  integral  function  of  a  certain  degree, 
having  a  certain  form  and  so  many  coefficients,  was 
to  them  as  much  an  unknown  quantity  as  the  pro- 
verbial x"1 

Happily  this  is  not  only  pedagogically  one  of  the 
first  steps,  but  practically  it  is  a  very  easy  one 
because  of  the  abundance  of  familiar  illustrations. 
"  Two  general  circumstances  strike  the  mind ;  one, 
that  all  that  we  see  is  subjected  to  continual  trans- 
formation, and  the  other  that  these  changes  are 
mutually  interdependent."2  V  Among  the  best  elemen- 
tary illustrations  are  those  involving  time;  a  stone 
falls,  and  the  distance  varies  as  the  time,  and  vice 
versa;  we  call  the  distance  a  function  of  the  time, 
and  the  time  a  function  of  the  distance.  We  take  a 
railway  journey;  the  distance  again  varies  as  the 
time,  and  again  time  and  distance  are  functions  of 
each  other.  Similarly,  the  interest  on  a  note  is  a 
function  of  the  time,  and  also  of  the  rate  and  the 
principal.,! 

This  notion  of  function  is  not  necessarily  foreign 
to  the  common  way  of  presenting  algebra,  except 
that  here  the  idea  is  emphasized  and  the  name  is 
made  prominent.  Teachers  always  give  to  beginners 
problems  of  this  nature :  Evaluate  x*  +  2  x  +  i  for 
x  =  2,  3,  etc.,  which  is  nothing  else  than  finding  the 

1  Presidential  address,  1885.  2  Laisant,  p.  46. 


ALGEBRA,  — WHAT  AND   WHY  TAUGHT  165 

value  of  a  function  for  various  values  of  the  variable. 
Similarly,  to  find  the  value  of  a?  +  3  cPb  +  3  aP  +.  & 
for  a  —  i,  b  =  2,  is  merely  to  evaluate  a  certain  func- 
tion of  a  and  b,  or,  as  the  mathematician  would  say, 
f(at  b),  for  special  values  of  the  variables.  It  is 
thus  seen  that  the  emphasizing  of  the  nature  of  the 
function  and  the  introduction  of  the  name  and  the 
symbol  are  not  at  all  diffioilt_for_beginners,  and  they 
constitute  a  natural  point  of  departure.  The  introduc- 
tion to  algebra  should  therefore  include  the  giving  of 
values  to  the  quantities  which  enter  into  a  function, 
and  thus  the  evaluation  of  the  function  itself. 

Having  now  defined  algebra  as  the  study  of  certain 
functions,1  which  includes  as  a  large  portion  the  solution 
of  equations,  the  question  arises  as  to  its  value  in  the 
curriculum. 

Why  studied  —  Why  should  one  study  this  theory  of 
certain  simple  functions,  or  seek  to  solve  the  quadratic 
equation,  or  concern  himself  with  the  highest  common 
factor  of  two  functions  ?  It  is  the  same  question  which 
meets  all  branches  of  learning, — cui  bono  ?  Why  should 
we  study  theology,  biology,  geology  —  God,  life,  earth  ? 
What  doth  it  profit  to  know  music,  to  appreciate  Pheid- 
ias,  to  stand  before  the  facade  at  Rheims,  or  to  wonder 

1  Certain  functions,  for  functions  are  classified  into  algebraic  and  trans- 
cendental, and  with  the  latter  elementary  algebra  concerns  itself  but  little. 
E.g.,  algebra  solves  the  algebraic  equation  x*  —  b,  but  with  the  transcen- 
dental equation  a*  =  b  it  does  not  directly  concern  itself. 


1 66    THE  TEACHING  OF  ELEMENTARY  MATHEMATICS 

at  the  magic  of  Titian's  coloring  ?  As  Malesherbes 
remarked  on  Bachet's  commentary  on  Diophantus,  "  It 
won't  lessen  the  price  of  bread ; " 1  or  as  D' Alembert 
retorts  from  the  mathematical  side,  a  propos  of  the  Iphi- 
genie  of  Racine,  "  What  does  this  prove  ? " 

Professor  Hudson  has  made  answer :  "  To  pursue  an 
intellectual  study  because  it  '  pays '  indicates  a  sordid 
spirit,  of  the  same  nature  as  that  of  Simon,  who  wanted 
to  purchase  with  money  the  power  of  an  apostle.  The 
real  reason  for  learning,  as  it  is  for  teaching  algebra,  is, 
that  it  is  a  part  of  Truth,  the  knowledge  of  which  is  its 
own  reward. 

"  Such  an  answer  is  rarely  satisfactory  to  the  ques- 
tioner. He  or  she  considers  it  too  vague  and  too  wide, 
as  it  may  be  used  to  justify  the  teaching  and  the  learn- 
ing of  any  and  every  branch  of  truth ;  and  so,  indeed,  it 
does.  A  true  education  should  seek  to  give  a  knowledge 
of  every  branch  of  truth,  slight  perhaps,  but  sound  as 
far  as  it  goes,  and  sufficient  to  enable  the  possessor  to 
sympathize  in  some  degree  with  those  whose  privilege  it 
is  to  acquire,  for  themselves  at  least,  and  it  may  be  for 
the  world  at  large,  a  fuller  arid  deeper  knowledge.  A 
person  who  is  wholly  ignorant  of  any  great  subject  of 
knowledge  is  like  one  who  is  born  without  a  limb,  and 
is  thereby  cut  off  from  many  of  the  pleasures  and  inter- 
ests of  life. 

1 "  Le  commentaire  de  Bachet  sur  Diophante  ne  fera  pas  diminuer  le 
prix  du  pain." 


ALGEBRA,  —  WHAT  AND  WHY  TAUGHT  167 

"  I  maintain,  therefore,  that  algebra  is  not  to  be  taught 
on  account  of  its  utility,  not  to  be  learnt  on  account  of 
any  benefit  which  may  be  supposed  to  be  got  from  it  ; 
but  because  it  is  a  part  of  mathematical  truth,  and  no 
one  ought  to  be  wholly  alien  from  that  important  depart- 
ment of  human  knowledge."  1 

The  sentiments  expressed  by  Professor  Hudson  will 
meet  the  '  approval  of  all  true  teachers.  Algebra  is 
taught  but  slightly  for  its  utilities  to  the  average  citizen. 
Useful  it  is,  and  that  to  a  great  degree,  in  all  subsequent 
mathematical  work  ;  but  for  the  merchant,  the  lawyer, 
the  mechanic,  it  is  of  slight  practical  value. 

Training  in  logic  —  But  Professor  Hudson  states,  in 
the  above  extract,  only  a  part  of  the  reason  for  teaching 
the  subject  —  that  we  need  to  know  of  it  as  a  branch  of 
human  knowledge.  This  might  permit,  and  sometimes 
seems  to  give  rise  to,  very  poor  teaching.  We  need  it 
also  as  an  exercise  in  logic,  and  this  gives  character  to 
the  teacher's  work,  raising  it  from  the  tedious,  barren, 
mechanical  humdrum  of  rule-imparting  to  the  plane  of 
true  education.  Professor  Hudson  expresses  this  idea 
later  in  his  paper  when  he  says,  "  Rules  are  always 
mischievous  so  long  as  they  are  necessary:  it  is  only 
when  they  are  superfluous  that  they  are  useful." 

Thus  to  be  able  to  extract  the  fourth  root  of  x*+4x* 
4*+  i  is  a  matter  of  very  little  moment.  The 


1  Hudson,  W.  H.  H.,  On  the  Teaching  of  Elementary  Algebra,  paper 
before  the  Educational  Society  (London),  Nov.  29,  1886. 


168    THE  TEACHING  OF  ELEMENTARY   MATHEMATICS 

pupil  cannot  use  the  result,  nor  will  he  be  liable  to  use 
the  process  in  his  subsequent  work  in  algebra.  But 
that  he  should  have  power  to  grasp  the  logic  involved 
in  extracting  this  root  is  very  important,  for  it  is  this 
very  mental  power,  with  its  attendant  habit  of  concen- 
tration, with  its  antagonism  to  wool-gathering,  that  we 
should  seek  to  foster.  To  have  a  rule  for  rinding  the 
highest  common  factor  of  three  functions  is  likewise  a 
matter  of  little  importance,  since  the  rule  will  soon  fade 
from  the  memory,  and  in  case  of  necessity  a  text-book 
can  easily  be  found  to  supply  it ;  but  to  follow  the  logic 
of  the  process,  to  keep  the  mind  intent  upon  the  opera- 
tion while  performing  it,  herein  lies  much  of  the  value 
of  the  subject,  —  here  is  to  be  sought  its  chief  raison 
d^tre. 

Hence  the  teacher  who  fails  to  emphasize  the  idea 
of  algebraic  function  fails  to  reach  the  pith  of  the 
science.  The  one  who  seeks  merely  the  answers  to 
a  set  of  unreal  problems,  usually  so  manufactured 
as  to  give  rational  results  alone,  instead  of  seeking  to 
give  that  power  which  is  the  chief  reason  for  alge- 
bra's being,  will  fail  of  success.  It  is  of  little  value 
in  itself  that  the  necessary  and  sufficient  condition 
for  x*  —  x  —  o  is  that  x  =  o,  ,r=  i^x  —  —  i ;  but  it  is 
of  great  value  to  see  why  this  is  such  condition. 

Practical  value  —  Although  for  most  people  algebra 
is  valuable  only  for  the  culture  which  it  brings,  at  the 
same  time  it  has  never  failed  to  appeal  to  the  common 


ALGEBRA,  —  WHAT  AND    WHY  TAUGHT  169 

sense  of  practical  men  as  valuable  for  other  reasons. 
All  subsequent  mathematics,  the  theory  of  astronomy, 
of  physics,  and  of  mechanics,  the  fashioning  of  guns, 
che  computations  of  ship  building,  of  bridge  building, 
and  of  engineering  in  general,  these  rest  upon  the  opera- 
tions of  elementary  algebra.  Napoleon,  who  was  not  a 
man  to  overrate  the  impractical,  thus  gave  a  statesman's 
estimate  of  the  science  of  which  algebra  is  a  corner- 
stone:  "The  advancement,  the  perfecting  of  mathe- 
matics, are  bound  up  with  the  prosperity  of  the  State." l 
Ethical  value — There  are  those  who  make  great  claims 
for  algebra,  as  for  other  mathematical  disciplines,  as 
a  means  of  cultivating  the  love  for  truth,  thus  giving 
to  the  subject  a  high  ethical  value.  Far  be  it  from 
teachers  of  the  science  to  gainsay  all  this,  or  to  antago- 
nize those  who  follow  Herbart  in  bending  all  education 
to  bear  upon  the  moral  building-up  of  the  child.  But 
we  do  well  not  to  be  extreme  in  our  claims  for  mathe- 
matics. Cauchy,  one  of  the  greatest  of  the  French 
mathematicians  of  the  nineteenth  century,  has  left  us 
some  advice  along  this  line:  "There  are  other  truths 
than  the  truths  of  algebra,  other  realities  than  those  of 
sensible  objects.  Let  us  cultivate  with  zeal  the  mathe- 
matical sciences,  without  seeking  to  extend  them  beyond 
their  own  limits;  and  let  us  not  imagine  that  we  can 
attack  history  by  formulae,  or  employ  the  theorems  of 

1  L'avancement,  le  perfectionnement  des  mathematiques  sont  lies  a  la 
prosperite  de  1'Etat. 


170    THE  TEACHING  OF  ELEMENTARY  MATHEMATICS 

algebra  and  the  integral  calculus  in  the  study  of  ethics." 
For  illustration,  one  has  but  to  read  Herbart's  Psychology 
to  see  how  absurd  the  extremes  to  which  even  a  great 
thinker  can  carry  the  applications  of  mathematics. 

Of  course  algebra  has  its  ethical  value,  as  has  every 
subject  whose  aim  is  the  search  for  truth.  But  the 
direct  application  of  the  study  to  the  life  we  live  is  very 
slight.  When  we  find  ourselves  making  great  claims 
of  this  kind  for  algebra,  it  is  well  to  recall  the  words 
of  Mme.  de  Stael,  paying  her  respects  to  those  who,  in 
her  day,  were  especially  clamorous  to  mathematicize  all 
life:  "Nothing  is  less  applicable  to  life  than  mathe- 
matical reasoning.  A  proposition  in  mathematics  is 
decidedly  false  or  true;  everywhere  else  the  true  is 
mixed  in  with  the  false." 

When  studied — Having  framed  a  tentative  defini- 
tion of  algebra,  and  having  considered  the  reason  for 
studying  the  science,  we  are  led  to  the  question  as 
to  the  place  of  algebra  in  the  curriculum. 

At  the  present  time,  in  America,  it  is  generally 
taken  up  in  the  ninth  school  year,  after  arithmetic 
and  before  demonstrative  geometry.  Since  most 
teachers  are  tied  to  a  particular  local  school  system, 
as  to  matters  of  curriculum,  the  question  is  not  to 
them  a  very  practical  one.  But  as  a  problem  of 
education  it  has  such  interest  as  to  deserve  attention. 

Quoting  again  from  Professor  Hudson :  "  The  be- 
ginnings of  all  the  great  divisions  of  knowledge 


ALGEBRA, -WHAT  AND   WHY  TAUGHT  171 

should  find  their  place  in  a  perfect  curriculum  of 
education ;  at  first  something  of  everything,  in  order 
later  to  learn  everything  of  something.  But  it  is 
needless  to  say  all  subjects  cannot  be  taught  at  once, 
all  cannot  be  learnt  at  once ;  there  is  an  order  to  be 
observed,  a  certain  sequence  is  necessary,  and  it  may 
well  be  that  one  sequence  is  more  beneficial  than  an- 
other. My  opinion  is  that,  of  this  ladder  of  learning, 
Algebra  should  form  one  of  the  lowest  rungs;  and  I 
find  that  in  the  Nineteenth  Century  for  October,  1886, 
the  Bishop  of  Carlisle,  Dr.  Harvey  Goodwin,  quotes 
Comte,  the  Positivist  Philosopher,  with  approval,  to 
the  same  effect. 

"  The  reason  is  this :  Algebra  is  a  certain  science, 
it  proceeds  from  unimpeachable  axioms,  and  its  con- 
clusions are  logically  developed  from  them ;  it  has  its 
own  special  difficulties,  but  they  are  not  those  of 
weighing  in  the  balance  conflicting  probable  evidence 
which  requires  the  stronger  powers  of  a  maturer 
mind.  It  is  possible  for  the  student  to  plant  each 
step  firmly  before  proceeding  to  the  next,  nothing  is 
left  hazy  or  in  doubt;  thus  it  strengthens  the  mind 
and  enables  it  better  to  master  studies  of  a  different 
nature  that  are  presented  to  it  later.  Mathematics 
give  power,  vigor,  strength,  to  the  mind ;  this  is 
commonly  given  as  the  reason  for  studying  them.  I 
give  it  as  the  reason  for  studying  Algebra  early,  that 
is  to  say,  for  beginning  to  study  it  early;  it  is  not 


172    THE  TEACHING  OF  ELEMENTARY   MATHEMATICS 

necessary,  it  is  not  even  possible,  to  finish  the  study 
of  Algebra  before  commencing  another.  On  the 
other  hand,  it  is  not  necessary  to  be  always  teaching 
Algebra ;  what  we  have  to  do,  as  elementary  teachers, 
is  to  guide  our  pupils  to  learn  enough  to  leave  the 
door  open  for  further  progress ;  we  take  them  over 
the  threshold,  but  not  into  the  innermost  sanctuary. 

"The  age  at  which  the  study  of  Algebra  should 
begin  differs  in  each  individual  case.  ...  It  must  be 
rare  that  a  child  younger  than  nine  years  of  age  is 
fit  to  begin;  and  although  the  subject,  like  most 
others,  may  be  taken  up  at  any  age,  there  is  no 
superior  limit;  my  own  opinion  is,  that  it  would  be 
seldom  advisable  to  defer  the  commencement  to  later 
than  twelve  years." 

This  opinion  has  been  quoted  not  for  indorsement, 
but  rather  as  that  of  a  teacher  and  a  mathematician 
of  such  prominence  as  to  command  respect.  The 
idea  is  quite  at  variance  with  the  American  custom 
of  beginning  at  about  the  age  of  fourteen  or  fifteen, 
or  even  later,  and  it  raises  a  serious  question  as  to 
the  wisdom  of  our  course.  Indeed,  not  only  is  the 
question  of  age  involved,  but  also  that  of  general 
sequence.  Are  we  wise  in  teaching  arithmetic  for 
eight  years,  dropping  it  and  taking  up  algebra,  drop- 
ping that  and  taking  up  geometry,  with  possibly  a 
brief  review  of  all  three  later,  at  the  close  of  the 
high  school  course? 


ALGEBRA,  — WHAT  AND  WHY  TAUGHT  173 

Fully  recognizing  the  folly  of  a  dogmatic  state- 
ment of  what  is  the  best  course,  and  hence  desiring 
to  avoid  any  such  statement,  the  author  does  not 
hesitate  to  express  his  personal  conviction  that  the 
present  plan  is  not  a  wisely  considered  one.  He 
feels  that  with  elementary  arithmetic  should  go,  as 
already  set  forth  in  Chapter  V,  the  simple  equation,1 
and  also  metrical  geometry  with  the  models  in  hand ; 
that  algebra  and  arithmetic  should  run  side  by  side 
during  the  eighth  and  ninth  years,  and  that  demon- 
strative geometry  should  run  side  by  side  with  the 
latter  part  of  algebra.  One  of  the  best  of  recent 
series  of  text-books,  Holzmtiller's,2  follows  this  general 
plan,  and  the  arrangement  has  abundant  justification 
in  most  of  the  Continental  programmes.  It  is  so  scien- 
tifically sound  that  it  must  soon  find  larger  acceptance 
in  English  and  American  schools. 

Arrangement  of  text-books  —  As  related  to  the  sub- 
ject just  discussed,  a  word  is  in  place  concerning  the 
arrangement  of  our  text-books.  It  is  probable  that 
we  shall  long  continue  our  present  general  plan  of 
having  a  book  on  arithmetic,  another  on  algebra,  and 
still  another  on  geometry,  thus  creating  a  mechanical 
barrier  between  these  sciences.  We  shall  also,  doubt- 

1  There  is  a  good  article  upon  this  by  Oberlehrer  Dr.  M.  Schuster,  Die 
Gleichung  in  der  Schule,  in  Hoffmann's  Zeilschrift,  XXlX.  Jahrg.  (1898), 
p.  Si. 

2  Leipzig,  B.  a  Teubner. 


174    THE  TEACHING  OF  ELEMENTARY  MATHEMATICS 

less,  combine  in  each  book  the  theory  and  the  exer- 
cises for  practice,  because  this  is  the  English  and 
American  custom,  giving  in  our  algebras  a  few  pages 
of  theory  followed  by  a  large  number  of  exercises. 
The  Continental  plan,  however,  inclines  decidedly 
toward  the  separation  of  the  book  of  exercises  from 
the  book  on  the  theory,  thus  allowing  frequent 
changes  of  the  former.  It  is  doubtful,  however,  if 
the  plan  will  find  any  favor  in  America,  its  advan- 
tages being  outweighed  by  certain  undesirable  fea- 
tures.1 There  is,  perhaps,  more  chance  for  the  adoption 
of  the  plan  of  incorporating  the  necessary  arithmetic, 
algebra,  and  geometry  for  two  or  three  grades  into 
a  single  book,  a  plan  followed  by  Holzmiiller  with 
much  success. 

1  An  interesting  set  of  statistics  with  respect  to  German  text-books  is 
given  by  J.  W.  A.  Young  in  Hoffmann's  Zeitschrift,  XXIX.  Jahrg.  (1898), 
p.  410,  under  the  title,  Zur  mathematischen  Lehrbiicherfrage. 


CHAPTER  VIII 
TYPICAL  PARTS  OF  ALGEBRA 

Outline  —  While  it  is  not  worth  while  in  a  work  of 
this  kind  to  enter  into  commonplace  explanations  of 
matters  which  every  text-book  makes  more  or  less 
lucid,  it  may  be  of  value  to  call  attention  to  certain 
topics  that  are  somewhat  neglected  by  the  ordinary 
run  of  classroom  manuals.  The  teacher  is  depend- 
ent upon  his  text-book  for  most  of  his  exercises, 
since  the  dictation  of  any  considerable  number  is  a 
waste  of  time.  He  is  likewise  dependent  upon  the 
book  for  much  of  the  theory,  since  economy  of  time 
and  of  students'  effort  requires  him  to  follow  the 
text  unless  there  is  some  unusual  reason  for  depart- 
ing from  it.  But  he  is  not  dependent  upon  the  book 
for  the  sequence  of  topics,  nor  for  all  of  the  theory, 
nor  for  all  of  his  problems;  neither  is  he  precluded 
from  creating  all  the  interest  possible,  and  introduc- 
ing a  flood  of  light,  through  his  superior  knowledge  of 
the  subject.  For  this  reason  this  chapter  is  written, 
that  it  may  add  to  the  teacher's  interest  by  throwing 
some  light  upon  a  few  typical  portions,  and  may 
suggest  thereby  some  improved  methods  of  treating 
the  entire  subject. 

175 


176     THE  TEACHING  OF  ELEMENTARY   MATHEMATICS 

Definitions  —  The  policy  of  learning  any  consider- 
able number  of  definitions  at  the  beginning  of  a  new 
subject  of  study  has  already  been  discussed  in  Chap- 
ter II.  The  idea  is  always  of  vastly  more  impop 
tance  than  the  memorized  statement.  At  the  same 
time  there  is  much  danger  from  the  inexact  defini- 
tions to  be  found  in  many  text-books,  a  danger  all 
the  greater  because  of  the  pretensions  of  the  science 
to  be  exact,  and  because  there  will  always  be  found 
teachers  who  believe  it  their  duty  to  burn  the  defini- 
tions indelibly  into  the  mind. 

Whether  the  definitions  are  learned  verbatim  or 
not,  the  teacher  at  least  will  need  to  know  whether 
they  are  correct.  For  this  purpose  he  will  find  little 
assistance  from  other  elementary  school-books.  He 
will  need  to  resort  to  such  works  as  Chrystal,1  as 
Oliver,  Wait,  and  Jones,2  or  as  Fisher  and  Schwatt3 
in  English,  as  Bourlet4  in  French,  as  the  convenient 
little  handbooks  of  the  Sammlung  Goschen6  or  the 
new  Sammlung  Schubert6  in  German,  and  Pincherle's 
little  Italian  handbooks.7 

1  Algebra,  2  vols.,  2  ed.,  Edinburgh,  1889. 

2  A  Treatise  on  Algebra,  Ithaca,  N.  Y.,  1887. 

3  Text-book  of  Algebra,  part  i,  Philadelphia,  1898. 

4  Lecons  d'Algebre  elementaire,  Paris,  1896. 

6  As  Schubert's  Arithmetik  und  Algebra,  and  Sporer's  Niedere  Analysis. 

6  As  Schubert's  Elementare  Arithmetik  und  Algebra,  and  Pund's  Alge- 
bra, Determinanten  und  elementare  Zahlentheorie,  both  published  in  1899. 

7  Algebra  elementare,  and  Algebra  complementare.     A  good  bibliog- 
raphy of  this  subject,  for  teachers,  is  given  by  T.  J.  McCormack  in  his 


TYPICAL  PARTS  OF  ALGEBRA  177 

A  few  illustrations  of  the  general  weakness  of  the 
common  run  of  definitions  may  be  of  service  in  the 
way  of  leading  teachers  to  a  more  critical  examina- 
tion of  such  statements. 

The  usual  definition  of  degree  of  a  monomial  is  so 
loosely  stated  that  the  beginner  thinks  and  continues 
to  think  of  3  a2*3  as  of  the  fifth  degree,  which  it  is 
in  a  and  x ;  but  for  the  purposes  of  algebra,  es- 
pecially in  dealing  with  equations,  it  is  quite  as  often 
considered  as  of  the  third  degree  in  x>  a  distinction 
usually  ignored  until  the  student,  after  much  stum- 
bling, comes  upon  it. 

A  square  root  is  usually  defined  as  one  of  the  two 
equal  factors  of  an  expression,  although  the  student 
is  taught,  almost  at  the  same  time,  that  the  expres- 
sion of  which  he  is  extracting  the  square  root  has 
no  two  equal  factors.  E.g.y  he  speaks  of  the  square 
root  of  x*  -h  i,  and  yet  says  that  x*  +  i  is  prime. 

Even  so  simple  a  concept  as  that  of  equation  is 
usually  defined  in  a  fashion  entirely  inexpressive  of 
the  present  algebraic  meaning.  Some  books  follow 
an  ancient  practice  of  avoiding  the  difficulty  by 
introducing  the  expression  "equation  of  condition," 
and  never  referring  to  it  again!  In  the  algebra  of 
to-day  an  equation  is  an  equality  which  exists  only 
for  particular  values  of  certain  letters  called  the 

notes  to  the  new  edition  of  De  Morgan's  work,  On  the  Study  of  Mathe- 
matics, Chicago,  1898,  p.  187. 


178    THE  TEACHING  OF  ELEMENTARY  MATHEMATICS 

unknown  quantities.  As  the  term  is  used  by  alge- 
braists of  the  present  time,  2  +  3  =  5  is  not  an  equa- 
tion strictly  speaking,  although  it  expresses  equality; 
neither  is  a*  -f  b  =  b  +  a2,  although  it  is  an  identity. 
An  equation,  as  the  word  is  now  used,  always  con- 
tains an  unknown  quantity.1 

The  term  "  axiom  "  is  subject  to  similar  abuse.  No 
mathematician  now  defines  it  as  "  a  self-evident  truth," 
and  no  psychology  would  now  sanction  such  an  unsci- 
entific statement.  Algebraists,  those  who  make  the 
science  to-day,  agree  that  an  axiom  is  merely  a  general 
statement  so  commonly  accepted  as  to  be  taken  for 
granted,  and  a  statement  which  needs  to  be  considered 
with  care  in  the  light  of  the  modern  advancement  of  the 
science.  For  example,  no  student  who  thinks  would 
say  that  it  is  "  self-evident "  that  "  like  roots  of  equals 
are  equal."  If  4  =  4,  it  is  not  "self-evident"  that  a 
square  root  of  4  equals  a  square  root  of  4,  for  +2  does 
not  equal  —  2. 

Again,  of  what  value  is  it  to  a  pupil  to  learn  the  ordi- 
nary definition  of  addition  ?  Text-books  commonly  say, 
in  substance,  that  the  process  of  uniting  two  or  more 
expressions  in  a  single  expression  is  called  addition ; 
but  what  is  meant  by  this  "  uniting  "  ?  Either  the  defi- 
nition would  better  be  omitted,  or  it  would  better  have 
some  approach  to  scientific  accuracy ;  the  choice  of 

1  De  Morgan's  use  of  the  word  is  not  that  of  modern  writers.  See  The 
Study  of  Mathematics,  2  ed.,  Chicago,  1898,  p.  57,  91. 


TYPICAL  PARTS  OF  ALGEBRA  179 

these  alternatives  may  depend  upon  the  class,  or  pos- 
sibly upon  the  teacher. 

The  simple  concept  of  factor,  so  vital  to  the  pupil's 
progress  in  algebra,  usually  suffers  with  the  rest.  Is  a 
factor,  as  we  so  often  read,  one  of  several  numbers  or 
expressions  which  multiplied  together  make  a  given 
expression  ?  In  other  words,  is  it  an  expression  which 
will  divide  another  ?  If  so,  are  Vr+  i  and  VJ—  i  fac- 
tors of  x—  i  ?  Possibly  it  will  be  said  that  we  are  limited 
to  rational  terms  in  x.  If  so,  when  we  ask  a  pupil  to 
factor  .z3—  i,  shall  we  expect  him  to  say  that  x*—  i  = 
(*-i)(jr+J  +  jV~^3)(;if+£-jV^"3)?  This  does  not 
involve  any  irrational  term  in  x.  But  possibly  we  are 
expected  to  exclude  irrational  and  imaginary  numbers 
altogether.  What,  then,  shall  we  say  about  factoring 
x*— \  ?  Are  the  factors  x+ \  and  x—  J,  or  are  fractions 
also  excluded  ?  Is  x2— a  factorable,  we  not  knowing  in 
advance  but  that  a =4  or  9  or  some  other  square? 
These  are  not  trivial  "  catch "  questions.  Upon  the 
answers  depends  the  entire  notion  of  factoring,  the 
basis  upon  which  we  are  to  build  the  greatest  part  of 
algebra  —  the  theory  of  equations. 

Of  less  importance,  but  still  of  value,  is  the  definition 
of  highest  common  factor.  What  is  the  highest  common 
factor  of  2  (a8- J8)  and  4 (£2- a2)?  Is  it  2(a-b\  or 
2(b  — a\  or  simply  ±(a— £)?  And  similarly,  what  is 
the  lowest  common  multiple  of  a  —  b  and  b—al  These 
questions  should  not  be  puzzling;  the  information  is 


ISO    THE  TEACHING  OF  ELEMENTARY   MATHEMATICS 

often  needed  in  the  simple  reduction  of  ordinary  frac- 
tions ;  and  yet  our  common  definitions  do  not  throw 
much  light  upon  them. 

The  unnecessary  and  ill-defined  term  "surd"  still  clings 
to  our  algebras.  Is  it  a  synonym  for  irrational  number  ? 
If  so,  what  is  an  irrational  number  ?  Is  it  a  number  not 
rational,  say  V2,  Va,  V— I  ?  Is  it  7r=3.i4i59---,  or 
the  circulate  0.666"-  ?  Is  it  a  single  expressed  root  like 
Vif,  or  is  2  +  V2  a  surd?  or  V2  +  V3?  or  \2  +  V$  ? 
If  it  is  merely  an  irrational  number,  is  log  2  a  surd  ? 
These  are  all  common  expressions,  arithmetical  rather 
than  algebraic,  it  is  true,  but  conventionally  holding  a 
place  in  algebra. 

In  this  connection  the  wonder  may  be  expressed 
as  to  how  long  we  shall  continue  to  use  the  terms 
"pure"  and  "affected"  (in  England  adfected)  quadrat- 
ics, instead  of  the  more  scientific  adjectives  "incom- 
plete" and  "complete." 

The  inquiry  might  be  extended  much  farther,  but 
enough  has  been  suggested  to  show  the  necessity 
for  care  in  the  common  definitions  of  algebra.1 

The  awakening  of  interest  in  the  subject,  the  vital 
point  in  all  teaching,  is  best  accomplished  through  the 

early  introduction   of   the   equation.     As   soon  as  the 

1 

1  For  those  who  have  not  access  to  the  works  mentioned  on  p.  176,  it 
may  be  of  service  to  refer  to  Beman  and  Smith's  Algebra,  Boston,  1900, 
in  which  the  authors  have  endeavored  to  state  the  necessary  definitions 
with  some  approach  to  scientific  accuracy. 


TYPICAL  PARTS  OF  ALGEBRA  l8l 

pupil  can  evaluate  a  few  functions,  thus  becoming 
familiar  with  the  alphabet  of  algebra,  the  equation 
should  be  introduced  with  this  object  prominently  in 
the  teacher's  mind. 

The  mere  solution  of  the  simple  equation  which 
the  pupil  first  meets  presents  no  difficulty.  The 
teacher  will  do  well  to  avoid  such  mechanical  phrases 
as  "clear  of  fractions"  and  "transpose"  until  the 
reasoning  is  mastered;  indeed,  it  may  be  questioned 
whether  these  phrases  are  ever  of  any  value.  Rather 
should  the  processes  stand  out  strongly,  thus:  — 

Given  -4-3  =  7,  to  nnd  ^te  value  of  x. 
2 

Subtracting  3  from  each  member,  -  =  4. 

2 

Multiplying  each  member  by  2,  x  =  8. 

To  prove  this  (check  the  result),  put  8  for  x\ 

then  -4-3  =  4+3  =  7. 

2 

But  the  greatest  difficulty  which  pupils  have  at 
this  time  comes  from  the  statement  of  the  conditions 
in  algebraic  language.  Fortunately  there  is  no  gen- 
eral method  of  stating  all  equations,  so  that  the  pupil 
is  forced  out  of  the  field  of  traditional  rules  into  that 
of  thought.  The  following  outline,  however,  is  usually 
of  value  in  arranging  the  statement :  — 

I.  What  shall  x  represent?  In  general,  x  may  be 
taken  to  represent  the  number  in  question.  E.g.,  in 


1 82    THE  TEACHING  OF  ELEMENTARY  MATHEMATICS 

the  problem,  "The  difference  of  two  numbers  is  40 
and  the  sum  is  50,  what  is  the  smaller  number  ? " 
Here  x  (or  some  other  such  symbol)  may  best  be 
taken  to  represent  "the  smaller  number." 

2.  For  what  member  described   in  the  problem  may 
two  expressions  be  found?    Thus  in  the   above  prob- 
lem, the  larger  number  is  evidently  50  —  xy  and  hence 
two  expressions  may  be  found  for  the  difference,  viz., 
40,  and  50  —  x  —  x. 

3.  How  do  you  state  the   equality  of  these   expres- 
sions in  algebraic  language  f 

$o-x-x=  40.1 

With  these  directions,  thus  outlining  a  logical  se- 
quence for  the  pupil,  the  statements  usually  offer 
little  difficulty. 

Signs  of  aggregation  often  trouble  a  pupil  more 
than  the  value  of  the  subject  warrants.  The  fact 
is,  in  mathematics  we  never  find  any  such  compli- 
cated concatenations  as  often  meet  the  student  almost 
on  the  threshold  of  algebra.  Nevertheless  the  sub- 
ject consumes  so  little  time  and  is  of  so  little  diffi- 
culty as  hardly  to  justify  any  serious  protest.  Two 
points  may,  however,  be  mentioned  as  typical. 

First,  it  is  a  waste  of  time,  and  often  a  serious 
waste,  to  require  classes  to  read  aloud  expressions  like 


1  Beman  and  Smith,  Algebra. 


TYPICAL  PARTS  OF  ALGEBRA  183 

There  is  no  value  in  such  an  exercise  in  oral  reading. 
Mathematicians,  if  by  strange  chance  they  should 
meet  such  an  array  of  symbols,  would  never  think 
of  reading  it  aloud.  Such  a  notion,  frittering  away 
time  and  energy  and  interest,  is  allied  to  that  which 
labors  to  have  —  a  called  "negative  a"  instead  of 
"minus  a,"  which  frets  about  "a  divided  by  b"  being 
called  "a  over  b"  (a  mathematical  expression  well 
recognized  by  the  best  writers  and  teachers  in  several 
languages),  and  which  objects  to  calling  a~n  "a  to 
the  minus  «th  power"  (forgetful  that  minus  and 
power  have  long  since  broadened  their  primitive 
meaning)  —  petty  nothings  born  of  the  narrow  views 
of  some  schoolmaster. 

The  second  point  refers  to  a  rule  which  still  finds 
place  in  many  text-books.  It  asserts  that  in  remov- 
ing parentheses  one  should  always  begin  with  the 
innermost,  proceeding  outward.  Consider,  for  exam- 
ple, these  solutions:  — 

Beginning  within  Beginning  without 


-(c-d-e)+c\        a- 
=a-[a+b-(c-d+e)+c\    =  a-a-b+(c-d-e)-c 


=  a  —  a  —  b+c—d+e—c         =za—a—b+c— 
=          —b-d+e  =          —b-d+c 

It  is   evident  that  there   are   fewer  changes  of   sign 
in  the  second  (4)  than  in  the  first  (8),  and  also  that 


1 84    THE  TEACHING  OF  ELEMENTARY   MATHEMATICS 

the  second  and  fourth  lines  in  the  second  could  have 
been  omitted  even  by  a  beginner.  The  only  excuse 
for  the  first  plan  is  that  it  affords  more  exercise;  but 
on  the  same  reasoning  a  child  would  do  well  to  per- 
form all  multiplications  by  addition. 

The  negative  number  is  supposed  to  be  the  first 
serious  crux  for  the  pupil  to  bear  in  his  journey 
through  algebra.  Much  has  been  written  as  to  the 
time  for  its  introduction.  Some  teachers  assert  that 
it  should  find  place  with  the  first  algebraic  concepts. 
Others  go  to  the  opposite  extreme  and  teach  the 
four  fundamental  processes  with  positive  integers,  and 
then  go  over  them  again  with  the  negative  number. 
Each  teacher,  like  each  text-book,  has  some  peculiar 
hobby,  and  rides  it  more  or  less  successfully.  As 
has  been  stated,  some  make  much  of  the  idea  that 
—  a  should  be  read  "  negative  a  "  instead  of  the  gen- 
erally recognized  "minus  a"  hoping  thereby  to  avoid 
the  confusion  thought  to  be  incident  to  the  two 
senses  in  which  "  minus "  is  used ;  others  (and  most 
of  the  world's  best  writers)  recognize  that  this  two- 
fold meaning  of  "  minus "  has  become  so  generally 
accepted  as  to  render  futile  any  attempt  at  change. 
The  very  diversity  of  view  shows  how  unimportant  is 
the  question  of  the  time  and  method  of  presenting 
the  subject,  and  of  the  language  in  question. 

The  writer  has  not  been  conscious  of  any  great 
difficulty  in  presenting  the  matter  to  classes,  and 


TYPICAL  PARTS  OF  ALGEBRA  18 


after  trying  the  various  sequences  has  for  some  time 
followed  this  plan  :  first  teach  a  working  knowledge 
of  the  alphabet  of  algebra,  through  the  evaluation  of 
simple  functions  ;  then  awaken  the  pupil's  interest  by 
the  introduction  of  some  easy  equations,  including  such 


as  V^+  2  =  8,  V;r  +1=3,  etc.;  then  show  the  neces- 
sity for  a  kind  of  number  not  commonly  met  in  arith- 
metic, developing  the  negative  number  and  the  zero. 

The  explanation  cannot  be  very  scientific  at  first. 
The  teacher  will  depend  largely  upon  graphic  illus- 
tration and  upon  matters  familiar  to  the  pupil.  The 
symbol  for  2°  below  zero,  for  50  years  before  Christ, 
the  symbols  for  opposite  latitudes  or  longitudes,  these 
lead  to  the  general  symbol  for  a  number  on  the  other 
side  of  a  zero  point  from  the  common  (positive) 
numbers.  The  ingenuity  of  teacher  and  pupils  then 
comes  into  play  in  the  way  of  illustrations  ;  the 
weight  of  a  balloon  when  empty,  when  full  of  gas; 
the  capital  of  a  man  who,  having  $5000,  loses  $3000, 
$5000,  £6000;  and  then  the  combined  weight  of  a 
10  Ib.  block  and  a  balloon  which  pulls  upward  with 
a  force  of  20  Ib.,  and  the  advantage  of  the  expression 
"  10  Ib.  and  minus  20  Ib." 

With  this  introduction  the  graphic  representation  of 
positive  and  negative  numbers  on  a  line  is  a  matter 
of  no  difficulty.  After  this  the  more  scientific  pro- 
cedure, showing  the  necessity  of  the  negative  number 
if  we  are  to  solve  an  equation  like  *  +  3  =  i,  and  the 


1 86    THE  TEACHING  OF  ELEMENTARY  MATHEMATICS 

definition  of  negative  numbers  and  of  absolute  values, 
complete  with  little  difficulty  the  elementary  theory. 

It  must  not  be  supposed  that  the  negative  number 
is  necessarily  approached  by  the  graphic  method. 
This  is  the  more  psychological,  but  not  the  more 
scientific  from  an  algebraic  standpoint.  Comte  long 
ago  pointed  this  out,  and  all  advanced  works  on  the 
theory  now  recognize  it.  "As  to  negative  numbers, 
which  have  given  rise  to  so  many  misplaced  discus- 
sions, as  irrational  as  useless,"  says  Comte,  "we  must 
distinguish  between  their  abstract  signification  and 
their  concrete  interpretation,  which  have  been  almost 
always  confounded  up  to  the  present  day.  Under 
the  first  point  of  view,  the  theory  of  negative  quan- 
tities can  be  established  in  a  complete  manner  by  a 
single  algebraical  consideration." l  It  is,  however, 
impossible  to  enter  into  any  extensive  discussion  of 
the  theory  at  this  time. 

1  Comte,  The  Philosophy  of  Mathematics,  translated  by  Gillespie,  N.  Y., 
1851,  p.  81. 

2  Most  teachers  have  access  to  Chrystal's  Algebra,  or  Fine's  Number 
System  of  Algebra,  and  these  works  give  satisfactory  discussions  of  the 
subject.       For  a  resume  of  the  matter  from  the  educational  standpoint 
it  is  well  to  read  the  Considerations  generates  sur  la  theorie  des  quan- 
tites  negatives,  et  objections  que  1'on  y  a   opposees,  in   Dauge's   Cours 
de  Methodologie  mathematique,  2.  ed.  p.  125.     But  the  best  works  for 
the  advanced  student  are  the  comparatively  recent  German  treatises  by 
Stolz,  Baltzer,  Biermann,  et  a/.,   or   Schubert's   Grundlagen    der    Arith- 
metik  in  the  Encyklopadie  der  mathematischen  Wissenschaften,  I.  Heft, 
Leipzig,  1898. 


TYPICAL  PARTS  OF  ALGEBRA  1 87 

Of  course  the  teacher  will  not  leave  the  subject 
without  having  the  pupil  understand  that  the  signs 
-f-  and  —  have  each  two  distinct  uses,  one  that  of 
symbols  of  operation,  as  in  10  —  8,  the  other  that  of 
quality,  as  —  8.  As  Cauchy  puts  it,  "  The  signs  + 
and  —  modify  the  quantity  before  which  they  are 
placed  as  the  adjective  modifies  the  noun."  Similarly, 
the  words  plus  and  minus  have  (as  noted  on  p.  184) 
two  distinct  uses,  as  in  "a  plus  quantity"  and  "a 
plus  b."  It  is  true  that  it  has  been  suggested  that  the 
expressions  "  plus  a "  and  "  plus  quantities "  should 
give  place  to  "  positive  a  "  and  "  positive  quantities," 
these  terms  being  more  precise.  But  much  as  we  may 
theorize  upon  the  desirability  of  such  usage,  the  fact 
remains  that  colloquially  the  shorter  expressions  are 
generally  used  by  the  world's  great  mathematicians, 
and  will  probably  continue  to  be  so  used. 

The  older  text-books  often  contain  a  great  deal  of 
worthless  matter,  and  worse,  about  proving  that  "  minus 
a  minus  is  plus,"  and  "minus  into  minus  is  plus,"  etc. 
Of  course  it  is  impossible  to  prove  any  such  thing  de 
novo.  Mathematicians  recognize  perfectly  well  that 
— a^  —  b=+ab  because  we  define  multiplication  involv- 
ing negatives  so  that  this  shall  be  true.  If  we  should 
change  the  definition  we  might  change  the  result  of  the 
multiplication.  All  that  is  to  be  expected  of  the  teacher 
is  that  it  should  be  shown  why  the  mathematical  world 
defines  —  a  •  —  b  to  mean  the  same  as  +a  •  +  bt  why  any 


1 88    THE  TEACHING  OF  ELEMENTARY  MATHEMATICS 

other  definition  would  be  inconsistent.  These  things 
are  easily  explained,  but  the  text-book  "  proofs "  of 
the  last  generation  have  now  been  discarded.  The 
favorite  one  of  these  "proofs"  was  this:  Since  multi- 
plying —  b  by  a  gives  —  ab,  therefore  if  the  sign  of 
the  multiplier  is  changed,  of  course  the  sign  of  the 
product  must  also  be  changed.  As  a  proof,  it  is  like 
saying  that  if  A,  a  white  man,  wears  black  shoes, 
therefore  it  follows  that  B,  a  black  man,  must  wear 
shoes  of  an  opposite  color. 

Checks — When  a  large  transatlantic  steamer  not 
long  since  ran  upon  the  rocks  near  Southampton, 
the  captain  announced  that  he  had  made  an  error  of 
a  few  miles  in  his  calculations.  Thousands  and 
thousands  of  dollars  lost,  hundreds  of  lives  jeopard- 
ized, just  because  a  simple  calculation  had  not  been 
checked!  And  yet  one  of  the  first  things  that  every 
computer  learns  is  the  necessity  for  checking  each 
operation,  a  necessity  which  should  be  impressed 
upon  the  student  of  algebra  from  the  first  day  of  his 
course.  It  is  a  matter  of  no  moment  whether  we  say 
"  check  "  or  "  prove  "  or  "  verify  "  ;  mathematicians 
probably  use  the  first  most  often ;  but  it  is  a  matter  of 
greatest  moment  that  we  see  that  each  step  is  right. 

What  checks  the  teacher  shall  require  depends 
somewhat  upon  the  pupils.  A  few  of  the  more  com- 
mon ones  will  be  suggested,  it  being  understood  that 
the  list  is  not  exhaustive. 


TYPICAL  PARTS  OF  ALGEBRA  189 

In  solving  an  equation  the  one  and  only  complete 
check  is  that  of  substituting  the  result  in  the  original 
equation  (in  the  statement  of  the  problem  if  there  be 
one).  It  makes  no  matter  what  axioms  we  use  or 
how  carefully  we  proceed;  a  result  is  right  if  it 
"checks,"  and  wrong  if  it  does  not.  As  Professor 
Chrystal  says :  "  The  ultimate  test  of  every  solution  is 
that  the  values  which  it  assigns  to  the  variables  shall 
satisfy  the  equations  when  substituted  therein.  No 
matter  how  elaborate  or  ingenious  the  process  by 
which  the  solution  has  been  obtained,  if  it  do  not 
stand  this  test,  it  is  no  solution ;  and,  on  the  other 
hand,  no  matter  how  simply  obtained,  provided  it 
do  stand  this  test,  it  is  a  solution."  1  Professor 
Henrici  expresses  the  same  thought  in  another  way : 
"  Simplifications  of  equations  follow  in  senseless  mo- 
notony, until  the  poor  fellow  really  thinks  that  solv- 
ing a  simple  equation  does  not  mean  the  finding  of  a 
certain  number  which  satisfies  the  equation,  but  the 
going  mechanically  through  a  certain  regular  process 
which  at  the  end  yields  some  number.  The  connec- 
tion of  that  number  with  the  original  equation  remains 
to  his  mind  somewhat  doubtful."  2 

To  illustrate,  consider  the  equation  x -1-2=3.  Sup- 
pose we  multiply  these  equals  by  x— 2,  the  results  must 
be  equal,  and  x*— 4=3^—  6,  whence  .z3— 3;r-f2=o. 

1  Algebra,  Vol.  I,  p.  286. 

3  Presidential  address,  Section  A,  British  Assn.,  1883. 


190    THE  TEACHING  OF  ELEMENTARY  MATHEMATICS 

Solving,  x—2,  or  i.  But  although  we  have  followed 
axioms  strictly,  #=  2  will  not  satisfy  the  original  equa- 
tion. So  with  any  equation,  the  pupil  who  checks  his 
work  is  master  of  the  situation ;  answer  books  are  only 
in  the  way,  save  in  the  case  of  unusually  complicated 
results,  and  the  pupil  knows  as  well  as  the  teacher  (per- 
haps better)  whether  his  result  is  right  or  wrong.  "  A 
habit  of  constant  verification  cannot  be  too  soon  encour- 
aged, and  the  earlier  it  is  acquired  the  more  swiftly  and 
almost  automatically  it  is  practised." 1 

A  very  useful  check,  applicable  to  the  operations  of 
algebra,  is  that  of  arbitrary  values.  Whatever  values 
are  assigned  to  a  and  b,  (a  +  frf  must  always  equal  a2  4- 
2  ab  +  ^.  In  other  words,  we  may  substitute  arbitrarily 
any  values  for  a  and  b,  and  see  if  the  two  forms  agree. 
E.g.,  let  #  =  2,  £=3;  then  (2 +  3)2=  22  +  2-2-3  +  32, 
which  is  true  because  each  is  25.  Or  suppose  a  pupil 
asserts  that  (x*  +  3*-  5)  O2  +  zx  —  i)  =  ^*  +  $*3  +  x* 
—  13*  +  5  ;  is  the  result  correct  ?  Substitute  any  arbi- 
trary value  for  x>  say  i,  and  the  question  reduces  to  this, 
Does  —  i  •  2  =  —  i  ?  Since  it  does  not,  there  is  evidently 
an  error.  The  arbitrary  value  i  is  usually  a  good  one 
unless  zero  enters  somewhere;  it  does  not  check  the 
exponents,  since  any  power  of  i  is  i,  but  mistakes 
are  not  usually  made  there.  Of  course  in  checking 
a  case  like  (x*  —  i)/(x  —  i)  =  x*  +  x  +  i,  it  will  not 

1  Heppel,  G.,  Algebra  in  Schools,  the  Mathematical  Gazette,  February, 
189$. 


TYPICAL  PARTS  OF  ALGEBRA  191 

do  to  use  the  value  i  for  x\  and  in  general  those 
values  should  be  avoided  which  make  any  expression 
zero. 

Another  check  extensively  used  by  mathematicians  is 
that  of  homogeneity.  The  name  is  long,  but  the  check 
is  simple.  "  At  present,  although  '  homogeneous '  is 
usually  defined  somewhere  in  the  first  three  pages  of 
a  school  algebra,  the  school-boy  never  knows  anything 
about  its  meaning,  as  he  has  not  been  used  to  apply  it." l 
The  check  simply  recognizes  the  fact  that  if  two  inte- 
gral functions  are  homogeneous,  their  sum,  difference, 
product,  and  powers,  are  homogeneous.  E.g.,  the  prod- 
uct of  a8  +  at?  and  a2  -f  ab  may  be  a5  +  aW  +  cfib  + 
a*&,  because  the  product  of  a  homogeneous  function  of 
the  third  degree  and  one  of  the  second  must  be  one  of 
the  fifth ;  but  if  the  result  is  given  as  a&  +  azP  +  cPb  + 
a2lP  there  must  be  an  error,  because  the  result  is  not 
homogeneous.  Since  homogeneous  functions  play  such 
an  important  part  in  mathematics,  this  check  is  of  more 
value  than  at  first  appears. 

Still  another  check,  less  extensively  used,  but  so 
easily  applied  as  to  be  valuable,  is  that  of  symmetry. 
If  two  functions  are  symmetric  with  respect  to  cer- 
tain letters,  their  product,  for  example,  must  be  sym- 
metric with  respect  to  those  letters.  E.g.,  x*—xy-\-y* 
and  x*+xy+y*  are  symmetric  with  respect  to  x  and 
y,  since  these  may  change  places  without  changing 

1  Heppel,  G.,  in  the  Mathematical  Gazette,  February,  1895. 


I  Q2     THE  TEACHING  OF  ELEMENTARY  MATHEMATICS 


the    forms    of    the    functions.      Hence 

may  be  their  product,  but  not  x*— 

although   it   checks   as   to   homogeneity   and    for    the 

arbitrary  values,  x=  i,  y—  i. 

The  first  two  of  the  checks  mentioned  should  be 
in  constant  use  by  the  student;  the  others  are  valu- 
able, but  not  indispensable. 

Factoring  has  already  been  mentioned  as  a  subject 
of  supreme  importance  in  algebra.  Pupils  waste 
much  time  in  performing  unnecessary  multiplications 
and  in  not  resorting  more  often  to  simple  factored 
forms.  For  example,  the  student  who  begins  the 
solution  of  the  equation 


by  clearing  of  fractions,  gets  into  trouble  both  theo- 
retically and  practically;  he  introduces  a  root  which 
does  not  belong  to  the  equation,  and  he  causes  him- 
self some  unnecessary  work.  He  should  see  at  a 
glance  that  x—  i  is  a  factor  of  2,xz  -f  3-tf2  —  4^—  i, 
and  can  easily  do  so  if  he  understands  the  elements 
of  the  subject. 

While  it  must  be  admitted  that  the  recent  text- 
books have  improved  upon  the  older  ones  in  the 
matter  of  factoring,  there  is  room  for  further  improve- 
ment. The  subject  is  often  divided  into  "cases," 
often  with  almost  no  difference,  as  with  x*  +  ax  +  £, 


TYPICAL  PARTS  OF  ALGEBRA  193 

x*  —  ax+b,  x*  +  ax  —  b,  etc.,  thus  leading  to  a  style 
of  treatment  that  is  depressing.  It  is  true  that  the 
arrangement  of  a  page  of  exercises  like  x2  +  ax  +  b, 
followed  by  another  of  the  type  x*  —  ax  +  b,  etc.,  has 
educational  value,  but  it  is  also  true  that  the  arrange- 
ment is  not  a  good  one.  It  reminds  one  of  the  six- 
teenth century  plan  of  having  one  rule  for  the  quad- 
ratic x*+px+q=o,  another  for  x2—Jur+g  =  o,  another 
for  x*  +  PX  —  q,  and  so  on.  The  favorite  answer  to 
all  this  is  that  pupils  cannot  generalize  and  take  the 
single  type  x2  -f  ax  4-  b,  where  a  or  b  may  be  either 
positive  or  negative;  but  the  experience  of  the  best 
teachers  shows  that  pupils  can  generalize  much  earlier 
than  some  of  our  text-books  would  seem  to  indicate. 
Some  special  forms  must  always  precede  the  general;  /  « 
but  to  give  only  special  forms,  never  referring  to  the 
general  type,  is  a  serious  error. 

The  fact  is,  there  are  only  a  few  distinct  types  of 
factored  expressions  that  are  of  much  value  in  subse- 
quent work.  The  most  important  are  (  i  )  ab  +  ac,  the 
type  involving  a  monomial  factor;  (2)  ax*+bx+c,  the 
general  trinomial  quadratic  in  x  ;  (3)  cases  involving 
binomial  factors  of  the  form  x—  a.  Of  course  for 
the  beginner  these  must  be  still  further  differenti- 
ated; but  problems  not  involving  these  three  cases, 
such  as  the  factoring  of 


and 


194    THE  TEACHING  OF  ELEMENTARY  MATHEMATICS 

have  value  rather  as  mental  gymnastics  than  as  cases 
to  be  used  in  subsequent  work. 

The  type  ax*-\-bx  +  c  includes  certain  special  cases 
which  must  be  considered  briefly  before  the  general 
one,  such  as  x*+2ax  +  a\  x*  —  a*y  x*  +  (a  +  b)x  +  ab, 
in  which  a  and  b  may  be  either  positive  or  negative. 
These  special  cases  are  satisfactorily  discussed  in 
most  text-books.  The  general  type,  ax*  4-  bx  +  c,  is 
not,  however,  so  well  treated.  There  are  numerous 
methods  of  attacking  it,  but  only  two  are  valuable 
enough  for  mention  here.  The  first  will  be  under- 
stood from  the  following  : 

17*+  12  =  6;tr2  4-9^+8*4-  12 


That  is,  the  17  is  separated  into  two  parts  whose 
product  is  6-12,  and  the  rest  of  the  work  is  simple. 
In  general,  in  ax*  +  bx  +  c,  the  b  is  separated  into 
two  parts  whose  product  is  ac.  The  reason  for  this 
is  easily  seen  by  considering  that 

(mx  4-  n)(m'x  +  n')=  mmrx2  +  (mnr  +  m'n)x  4-  nn*  ; 


that  is,  that  the  coefficient  of  x  is  made  up   of   two 
parts,  mn'  and  m'n,  whose  product  is  mm*  •  nn*. 

The  other  plan  consists  in  making  the  coefficient 
of  x*  a  square,  thus  : 

-  i?x+  12  =  ^(36^4-  17-6*4-72). 


TYPICAL  PARTS  OF  ALGEBRA  195 

Now  let  z  =  6x,  and  we  have 


Which  of  these  plans  is  followed  is  immaterial,  the 
rationale  of  each  being  easily  explained.  But  it  is 
needless  to  say  that  the  cut-and-  try  method  often  given, 
of  taking  all  possible  factors  of  6  x*  and  of  1  2  and  guess- 
ing at  the  proper  combination,  has  little  to  recommend  it. 

The  cases  involving  binomial  factors  of  the  form 
x  —  a  are  perhaps  the  most  important  of  any  which 
the  pupil  meets  in  his  elementary  work,  since  they 
enter  so  extensively  into  the  theory  of  equations. 
They  are  best  treated  by  the  remainder  theorem, 
which  has  long  found  place  in  the  closing  pages  of 
many  advanced  algebras,  where  it  could  not  be  used 
to  any  extent.  The  theorem  asserts  that  the  remain- 
der arising  from  dividing  an  integral  function  of  x 
by  x  —  a  can  be  found  in  advance  by  putting  a  for 
x  in  the  given  function.  E.g>,  in  dividing  x*  —  x* 
4-5^—  i6x+  n  \>y  x  —  I  we  know  that  there  will 
be  no  remainder,  for  1  —  1  +  5  —  16+11=0;  but  if 
x  —  2  is  the  divisor,  there  will  be  a  remainder  7,  for 

24  —  23+5«22—  l6'2+II  =  l6  —  8  +  2O  —  32+11  =  7. 

Similarly,  x11  —  y11  is   divisible   by  x—  y,  for  if  y   is 
put  for  x,  x11  —  y11  =y7  —  j/17  =  o;  but  it  is  not  divisi- 


196    THE  TEACHING  OF  ELEMENTARY   MATHEMATICS 

ble  by  x+y,  i.e.,  by  x  —  (— y\  for  if  —  y  is  put  for  x, 
(  -  j)17  -  y 7  =  -  y 7  -  717  =  -  2 y 7,  the  remainder.  The 
theorem  is  easily  proved,  and  its  usefulness  in  ele- 
mentary algebra  can  hardly  be  overestimated.  The 
proof,  condensed  more  than  advisable  for  beginners, 
is  as  follows : 

Let  /  (x)  be  the  dividend,  x  —  a  the  divisor,  q  the 
quotient,  r  the  remainder. 

Then  f(x)  =  (x  —  a)q  +  rt  in  which  r  cannot  con- 
tain x. 

This  being  an  identity  is  true  for  all  values  of  x, 
and  hence  for  x—  a. 

But  if  a  is  put  for  x\  we  have  f  (a)  =  r. 

I.e.,  the  remainder  is  the  same  as  f(x)  with  a  put 
for  x. 

A  teacher  will  have  no  difficulty  in  putting  this 
into  a  form  easily  comprehended  by  beginners.  The 
theory  is  not  difficult,  and  the  practice  is  very 
simple. 

It  is  unfortunate  that,  having  spent  considerable 
time  upon  the  subject  of  factoring,  so  many  text- 
books thereupon  relegate  it  to  the  mathematical 
garret.  The  next  chapter  is  usually  upon  highest 
common  factor,  in  which  the  pupil  is  led  to  make 
as  little  use  of  factoring  as  possible !  After  con- 
sidering the  lowest  common  multiple,  the  text-books 
next  proceed  to  fractions,  and  here  the  pupil  is 
led  to  use  the  highest  common  factor  in  his  reduc- 


TYPICAL  PARTS  OF  ALGEBRA  197 

tions,  which  we  rarely  do  in  practice,  but  other- 
wise the  important  subject  of  factoring  sinks  into 
disuse. 

What  is  the  remedy  for  this  evil?  The  answer 
appears  when  we  consider  the  common  uses  to  which 
the  mathematician  puts  the  subject.  He  has  two 
uses  for  it,  the  first  being  in  the  solution  of  equa- 
tions, and  the  second  in  shortening  his  work,  as  in 
the  reduction  of  fractions  to  forms  more  easily  han- 
dled. Hence  it  is  proper  to  follow  factoring  at  once 
with  some  simple  work  in  equations,  and  as  soon  as 
fractions  are  met  to  use  factoring  in  all  simple  re- 
ductions, reserving  the  highest  common  factor  for 
cases  of  real  difficulty. 

The  application  of  factoring  to  the  solution  of 
equations  is  very  simple,  if  the  pupil  knows  what 
it  means  to  solve  an  equation  like 


namely,  to  find  a  value  of  x  which  shall  make  the 
first  member  zero.  That  is,  the  value  of  x  which 
makes  x  —  a  =  o  is  evidently  a.  The  values  which 
make  ^  —  3^  —  4  =  0,  or,  what  is  the  same  thing, 
(x  —  £)(x  +  i)  =  o,  are  evidently  4  and  —  i,  because 
if  .#  =  4  we  have  0-5=0,  and  if  x=  —  i  we  have 
—  5-0  =  0.  Similarly,  the  values  which  make 

X*  —  a*x  =  o,  or  x(x  +  a)  (x  —  a)  =  o, 


198    THE  TEACHING  OF  ELEMENTARY  MATHEMATICS 

are  evidently  o,  —  a,  +  a.  In  this  way  a  consider- 
able number  of  equations  with  commensurable  roots 
should  be  given,  together  with  problems  involving 
equations  of  degree  above  the  first,  thus  at  the  same 
time  adding  to  the  interest  in  the  subject,  giving  drill 
in  factoring,  and  laying  a  rational  foundation  for 
quadratics.  A  pupil  so  trained  would  not,  on  reach- 
ing the  chapter  on  quadratics,  waste  his  time  "com- 
pleting the  square"  in  the  solution  of  such  equations 
as  x*  -f-  2  x  =  o,  or  x2  +  5  x  +  6  =  o.  It  takes  but  little 
time  to  introduce  this  work,  whatever  text-book  is  in 
use,  and  !he  benefit  derived  is  evident. 

In  the  treatment  of  fractions,  to  apply  the  Eucli- 
dean method  of  highest  common  factor  1  to  the  reduc- 
tion of  forms  like 


and 


*  +  9-r+  14  x  +    *   +  5*-  14 

is  to  encourage  the  pupil  to  waste  time  and  to  forget 
his  elementary  work  in  factoring. 

The   quadratic  equation,  often   looked  upon  as  the 
final  chapter  of  elementary  algebra,  seems  peculiarly 
open  to  mechanical  treatment.     Add  the  square  of  half 
the  coefficient  of  x,  extract  the  square  root,  transpose  — 
this  is  the  rule  ;  the  validity  of  the  result  is  not  consid- 

1  "Then  there  are  processes,  like  the  finding  of  the  G.  C.  M.,  which 
most  boys  never  have  any  opportunity  of  using,  excepting  perhaps  in  the 
examination  room."  Henrici,  Presidential  address,  British  Association, 
Section  A,  1883. 


TYPICAL  PARTS  OF  ALGEBRA  199 

ered  essential.  The  reason  for  this  procedure  is  doubt- 
less historical ;  the  early  mathematicians  were  forced  to 
solve  in  this  way,  and  the  tradition  has  endured  to 
the  present. 

But  if  we  are  to  follow  this  mechanical  route,  we  may 
well  go  even  farther.  For  practical  purposes  the  pupil 
eventually  needs  to.  be  able  to  write  down  at  sight  the 
roots  of  equations  like  x?+2x+$  =  ot  without  stop- 
ping to  "complete  the  square";  for  this  purpose  the 
formula 

^.^Viyarris 

22^  ^ 

should  be  as  familiar  to  him  as  the  multiplication  table. 
To  use  the  method  of  the  completion  of  the  square  in  a 
thoughtless  way  with  every  equation  has  neither  a  cul- 
ture value  (since  the  logic  is  concealed)  nor  a  utilitarian 
value  (since  it  is  an  unnecessarily  tedious  way  of  reach- 
ing the  result). 

The  best  plan  of  attacking  the  quadratic  equation  is, 
as  already  intimated,  through  factoring.  The  plan  is 
simple,  it  is  general  (not  being  limited  to  quadratics), 
it  can  be  introduced  with  factoring  and  continually 
reviewed  until  the  chapter  on  quadratics  is  reached,  and 
at  the  same  time  it  keeps  the  subject  of  factoring  fresh 
in  mind.  When  the  chapter  on  quadratics  is  reached, 
the  student  is  already  able  to  handle  the  ordinary  run 
of  manufactured  problems,  those  which  "come  out 
even  " —  with  small  integers  for  roots.  Those  involving 


20O    THE  TEACHING  OF   ELEMENTARY   MATHEMATICS 

large  numbers,  however,  require  other  methods,  and 
this  leads  to  the  completion  of  the  square,  an  expression 
derived  from  the  old  geometric  method  of  solving  the 
quadratic  equation.  The  outcome  of  this  method  should 
be  the  proof  of  the  fact  that  if 


-4?. 


or,  if  preferred,  the  formula  for  solving  ax*  +  bx  -f  c  =  o. 
This  formula,  logically  developed,  is  so  important  as  to 
demand  sufficient  application  to  fix  it  in  mind  for  use  in 
the  subsequent  parts  of  algebra.  That  a  pupil  should 
"  complete  the  square  "  every  time  he  runs  against  an 
equation  like  x*  +  x  4-  i  =  o  is  as  senseless  as  to  require 
him  to  add  three  I3's  when  he  wishes  the  product  of  3 
and  13. 

Some  text-books  give  one  or  two  other  methods  of 
solving  the  quadratic,  but  these  serve  to  confuse  rather 
than  assist  the  pupil.  Their  interest  is  more  historical 
than  educational.  That  the  teacher  may  see  that  the 
standard  solution  is  not  the  only  one,  however,  a  few 
historical  devices  may  be  of  service  : 

Method  of  Brahmagupta  (b.  598)  and  Bhaskara 
(b.  HI4).1 

Given  ax*  +  bx—  c. 

Then  4  d*x*  +  ^abx  =  4  ac9 


1  Matthiessen,  Grundziige  der  antiken  und  modernen  Algebra,  2.  Ausg., 
Leipzig,  1896,  p.  282. 


S    w  '  rX 

Of  T  M  «  X 

^  UNIVERSITY  ) 


TYPICAL  PARTS  OF  ALGEBRA  2OI 

-h  4  tf  for  -f  £*  =  4  tf£  + 


This  plan,  here  given  in  complete  form  with  modern 
symbols,  is  sometimes  called  the  Hindu  method.  It  has 
the  advantage  of  avoiding  fractions  until  the  last  step. 
Method  of  Mohammed  ben  Musa  (about  800,  see 
p.  151)  and  Omar  Khayyam  (d.  1123,  the  author  of 
the  Rubaiyat),  one  of  several  given  by  them,  and 
based  on  geometric  considerations.1 

Given  x*  —  px  =  g. 


P          f  P\ 

••>+?-(*-£) 

and 


or 


This  plan  is  essentially  the  one  now  in  general  use. 
Method  of  Vieta  (i6is).a 

Given  x*  -f  ax  -f-  b  =  o. 

Let  x  =  */  +  2. 

Then     ^P  -f  (22  +  d)u  +  (z*  +  az  +  b)  =  o. 

x  Matthiessen,  p.  309.  2  Matthiessen,  p.  311. 


202    THE  TEACHING  OF  ELEMENTARY  MATHEMATICS 

Since  but  one  condition  has  been  placed  upon  u  +  z9 
we  may  impose  another,  and  let 


whence 
and 


whence        x-=u-\- z  —  —  \a±\  \a2  —  4 b. 

Here  there  has  been  no  "completion  of  the  square." 
Method  of  Grunert  (186s).1 

Given  x*  +  ax  +  b  =  o. 

Let  x  =  u  +  z. 

But       (u  4-  ^)2  -  2  &  0  -f  z)  +  (^2  -  £2)  =  o. 

.-.  «  =  -2,  and^  =  ±iV^2-4^, 

from  which  ^r  is  easily  found. 

Fischer's  trigonometric  method  (1856)  is  one  of  sev- 
eral of  this  class. 

Given  x1  —  px  +  q  =  o,  with  />2  >  4  q. 

Let  #!  =  /  •  cos2  </>,  one  root, 

and  x<i  =/  •  sin2  <£,  the  other. 

Then     4rx  +  x^  =  /  (cos2  <£  -f-  sin2  <^>)  =  /, 
and  ^  ^r0  =  *>2  (sin  cf>-  cos  <f>^2  =  i  ^2-  sin2  2  (6. 


But  ^^2  =  ^,   /.  sin  2<£  = 

1  Grunert's  Archiv,  Bd.  40. 


TYPICAL  PARTS  OF  ALGEBRA  203 

For  example,  to  solve  x?  —  93. 7062  ;r  +  1984.74  =  o. 
Here     2<f>  =  71°  5/44."6,     /.<#>  =  35°  &  52."3, 
whence  ^  =  61.3607,   ^3  =  32-3454- 

The  problem  shows  that  trigonometry  is  able  materially 
to  assist  in  the  solution  of  certain  kinds  of  quadratic 
equations. 

There  are  many  other  devices  for  solving  the  quad- 
ratic, for  which  the  reader  must,  however,  be  referred 
to  the  great  compendium  of  Matth lessen.  Enough  of 
these  plans  have  been  suggested  to  show  that  a  de- 
parture from  the  single  one  in  general  use,  for  the 
purpose  of  emphasizing  the  method  of  factoring  and 
the  use  of  the  formula,  is  not  a  novelty  to  be  feared; 
it  is  merely  to  make  a  judicious  selection  from  the 
abundance  of  material  at  hand. 

Equivalent  equations  —  To  the  student  who  has  not 
been  taught  that  there  is  no  escape  from  the  check- 
ing of  the  roots  of  an  equation,  and  that  extraneous 
roots  are  liable  to  enter  with  any  one  of  several  com- 
mon operations,  it  seems  sufficient  to  blindly  follow 
the  axioms  until  a  solution  is  reached.  But  this 
is  so  far  from  the  case,  and  the  text-books  offer  so 
little  upon  the  subject,  that  a  brief  statement  con- 
cerning the  matter  may  be  of  service  to  teachers. 

While  it  is  true  that  the  solutions  of  equations  de- 
pend upon  a  few  well-known  axioms,  these  axioms 
may  lead  the  student  into  difficulty.  For  example: 


2O4    THE  TEACHING  OF  ELEMENTARY  MATHEMATICS 

Let  x  —  a. 

Then,  multiplying  by  x,        x2  =  ax. 
Subtracting  a*,  x*  —  a2  =  ax  —  a2. 

Factoring,   -        (x  +  a)t(x  —  a)  =  a (x  —  a). 
B\itx=at  .'.  2a(x—  a)  =  a(x  —  a). 

Dividing  by  x.  —  a, 

2  a  =  a,  or  2=1. 

Here  every  step  follows  from  the  preceding  one  by 
the  application  of  a  common  axiom,  ^and  yet  the 
result  is  absurd. 

Pupils  are  apt  to  place  undue  weight  upon  ,demon- 
strations  apparently  valid  but  in  reality  fallacious. 
But  as  J.  Bertrand,  the  French  algebraist,  says,  "  Com- 
mon sense  never  loses  its  rights ;  •  to  set  up  against 
evidence  a  demonstrated  formula  is  about  like  telling 
a  man  that  he  is  dead  because  you  happen  to  have 
a  physician's  certificate  to  that  effect." 

This  tendency  of  pupils  and  this  testimony  of  M. 
Bertrand  suggest  the  question :  What  limitations  are 
there  on  the  use  of  the  axioms  ?  To  answer  this 
question  requires  the  definition  of  equivalent  equa- 
tions. Two  equations  are  said  to  be  equivalent  when 
.all  of  the  roots  of  either  are  roots  of  the  other. 
E.g.,  x'+  3=  $x  —  i  and  x  +  i  =  3 (x—  i)  are  equiva- 
lent equations,  for  x  =  2  is  a  root,  and  the  only  root, 
of  each.  But  x  =  3  and  x*  =  9  are  not  equivalent, 


TYPICAL  PARTS  OF  ALGEBRA  2O5 

f or  x  =  —  3  is  one  root  of  the  second,  but  it  is  not  a 
root  of  the  first. 

It  is  axiomatic  that  if  equals  are  added  to  equals 
the  results  are  equal,  but  it  does  not  follow  that  the 
resulting  equation  is  equivalent  to  either  of  the  orig- 
inal ones.  E.g. : 

If  *=i, 

then  **=  i. 

Adding,  x*+x=2. 

Solving,  x  =  I  or  —  2. 

The  —  2  is  a  root  of  x*  +  x  =  2,  but  not  of  x  =  I 
nor  of  ;r2=  I.  The  equation  x2  +  x=2  is  not  equiva- 
lent to  either  of  the  others. 

It  is  also  an  axiom  that  if  equals  are  multiplied  by 
equals  the  results  are  equal.  But  it  does  not  follow 
that  the  resulting  equation  is  equivalent  to  the  others. 
E.g.,  if  x—  1  =  1,  and  we  multiply  by  x+  i,  while 
it  is  true  that  x*  —  i=x+i,  it  does  not  follow  that 
its  roots  are  the  same  as  that  of  x—  i  =  i.  They 
are  not,  for  x^—\—x-\-\  has  two  roots,  2  and  —  i, 
but  —  i  is  not  a  root  of  the  first  equation.  And  in 
general,  if  we  multiply  by  a  function  of  x  we  intro- 
duce (if  the  equation  is  integral)  one  or  more  new 
roots,  "extraneous  roots"  as  they  are  called. 

Similarly,  if  x  =  5,  then  x*  =  25,  x*=  125,  **  =  625,.-.; 
but  the  second  equation  has  one  root  which  the  first 


2O6    THE  TEACHING  OF  ELEMENTARY  MATHEMATICS 

has  not,  —  5 ;  the  third  has  two  which  the  first  has 
not,  5  (  —  ^  ±  JV  —  3) ;  the  fourth  has  three  extra- 
neous to  the  first,  and  so  on. 

Furthermore,  the  axiom  of  dividing  equals  by 
quals  needs  watching.  If  x*  +  2  x*  —  x  —  o,  then  by 
dividing  by  x,  x*  +  2  x  —  i  =  o,  or  x  —  —  i  ±  V~2. 
These  are  roots  of  the  original  equation,  but  they 
are  not  the  only  ones ;  x  —  o  is  also  a  root.  And  in 
general,  dividing  by  a  function  of  x  loses  one  or 
more  roots. 

In  dealing  with  radical  equations  the  difficulty  is 
even  more  pronounced.  When  we  deal  with  radicals 
it  is  customary  to  consider  only  the  sign  expressed 
before  them,  or  if  none  is  expressed  to  understand 
the  plus  sign.  That  is,  we  consider  the  value  of 
V4  +  VQ  to  be  2  +  3=5,  an<l  not  (±  2)  +  (±  3)  =  5, 
—  i,  i,  or  —5.  This  is  purely  conventional;  it  has 
simply  been  agreed  that  in  elementary  work  the  stu- 
dent shall  not  be  bothered  with  the  ±  unless  it  is 
expressed,  as  ±Vr4±A/9=  5,  —  i,  i,  —  5.  Since  the 
square  root  of  4  is  either  2  or  —  2,  it  is  evident  that 
the  plan  is  not  very  scientific;  but  so  long  as  it  is 
understood  no  great  harm  can  come  from  it.  So  when 
we  are  dealing  with  the  radical  equation  V^r—  i  =  3,  we 
seek  the  root  which  satisfies  the  equation  +V^r—  i 
=  3  and  not  —  V^r  —1  =  3,  although  of  course  the 
square  root  of  x—  I  is  both  plus  and  minus.  With 
this  understanding,  consider  the  following  solution : 


;tr—  c=i+;r— 2  —  2  V  X  —  2. 


TYPICAL  PARTS  OF  ALGEBRA 

Given 
Squaring, 
Hence 
and  x  =  6. 

But  on  substituting  6  for  x,  we  have 

VI  =  i  —  V^, 
or  1  =  1  —  2. 

That  is,  if  we  understand  VI  to  mean  the  positive 
square  root  of  i  and  not  the  negative  one,  6  is  not 
a  root.  It  is  therefore  called  extraneous,  and  the 
equation  is  said  to  be  insoluble.  However  unscien- 
tific this  may  seem,  the  limitation  of  the  sign  before 
the  radicals  in  such  way  as  to  make  many  equations 
insoluble,  it  has  high  mathematical  sanction.1  At  any 
rate,  it  is  evident  that  the  application  of  the  axioms 
gives  rise  to  roots  commonly  considered  extraneous. 

Considerations  such  as  these  show  how  necessary 
it  is  to  make  more  of  the  logic  of  algebra  than  is 
usually  done.  The  average  pupil  in  algebra  seems 
quite  content  if  able  to  say,  "Transposing  I  got  this, 
and  by  squaring  I  got  this,  and  the  next  step  came 

1 "  Following  Chrystal,  Todhunter,  Hall  and  Knight,  and  the  majority 
of  writers,  Va  should  be  considered  a  quantity  having  one  and  not  two 
values,  although  the  algebra  of  C.  Smith  and  the  article  by  Professor 
Kelland  in  the  Encyclopaedia  Britannica  make  Va  have  two  values." 
G.  Heppel  in  the  Mathematical  Gazette,  February,  1895. 


208    THE  TEACHING  OF  ELEMENTARY   MATHEMATICS 

from  dividing,"  etc.,  with  no  thought  as  to  the  legiti- 
macy of  the  process.  He  gives  with  each  step  the 
"  How,"  and  teachers  are  often  content ;  but  this  is 
of  relatively  minor  importance,  the  great  questions  to 
be  asked  at  each  step  being,  "  Why  is  this  true  ? " 
and,  "  Is  the  process  reversible  ?  " 
,/  Simultaneous  equations  and  graphs — There  is  often 
an  objection  raised  against  the  introduction  of  graphs 
in  elementary  algebra,*  that  there  is  no  reason  for 
thus  anticipating  analytic  geometry.  We  are  told 
that  algebra  and  geometry  are  separate  sciences, 
although  'this  separation  is  really  a  recent  event  in 
the  history  of  the  two  subjects.  What  a  striking 
little  rebuke  to  those  who  would  build  impassable 
barriers  between  the  branches  of  mathematics  which 
we  vainly  try  to  separate  by  distinctive  names,  is 
the  epigram  of  Sophie  Germain,  "Algebra  is  only 
written  geometry  —  geometry  merely  pictured  alge- 
bra ! " l  The  introduction  of  the  graph  is  so  simple, 
and  throws  such  a  flood  of  light  upon  simultaneous 
equations,  that  teachers  who*  have  used  the  plan 
rarely  abandon  it.  A  pupil  can  understand  much 
more  fully  why  two  linear  equations  with  two  un- 
knowns are  in  general  simultaneous,  if  the  matter  is 
brought  to  the  eye,  by  the  two  lines  which  represent 

1  L'algebre  n'est  qu'une  geometric  ecrite,  la  geometric  n'est  qu'une 
algebre  figuree.  It  recalls  Goethe's  description  of  architecture  as  "  frozen 
music,"  eine  erstarrte  Musik,  which  struck  Mme.  de  Stae'l  as  .so  felicitous. 


TYPICAL  PARTS  OF  ALGEBRA  209 

these  equations,  than  he  can  by  an  analytic  proof. 
He  sees,  too,  why  the  attempt  to  solve  the  set 

2*  +  67  =5,  3*  +  97  =7, 

fails.  If  he  is  told  that  in  general  three  linear  equa- 
tions are  not  simultaneous,  the  'reason  is  more  clear 
when  supplemented  by  the  pictures,  the  graphs,  of 
the  equations.  When  he  finds  that  in  spite  of  the 
general  fact  just  stated,  the  special  equations 


are    simultaneous,   and    that,    indeed,   others    can    be 
added  to  the  set,  as 

47*+r37=26,  15*+  157=30,  etc., 

the  mystery  of  the  matter  vanishes  as  soon   as  the 
graphs  are  plotted. 

Similarly  for  an  equation  of  the  second  degree  com- 
bined with  another  of  that  degree  or  with  a  linear 
equation.  While  there  is  a  simple  proof  that  in  gen- 
eral two  simultaneous  quadratics  cannot  be  solved 
without  recourse  to  a  quartic  equation,  most  students 
fail  to  appreciate  the  fact  until  they  have  the  assist- 
ance of  graphs.  Most  pupils  who  have  "finished" 
quadratics  would  expect  to  be  able  to  solve  the  set 

5*  +  67  +  7  =  0, 
—  17  y  —  2O  =  O, 


210    THE  TEACHING  OF  ELEMENTARY  MATHEMATICS 

and  would  wonder  at  their  inability  to  handle  it. 
They  cannot  understand  why  such  an  innocent  look- 

ing set  as 

xz+y  =  7,  x+y*  =  ii, 

(partly  soluble  by  quadratics  if  one  makes  the  lucky 
hit)  should  give  them  trouble.  They  are  satisfied 
with  one  or  two  roots  of  the  set 


-7  =  0,  x*  +  $xy  +  2y*  -  8  =  o, 

or  with  half  a  dozen,  if  by  the  introduction  of  extra- 
neous ones  they  can  get  together  that  number.  "  The 
curious  thing  is  that  many  examination  candidates 
who  show  great  facility  in  reducing  exceptional  equa- 
tions to  quadratics  appear  not  to  have  the  remotest 
idea  beforehand  of  the  number  of  solutions  to  be 
expected  !  and  that  they  will  very  often  produce  for 
you  by  some  fallacious  mechanical  process  a  solution 
which  is  none  at  all."1 

A  valuable  exercise  for  a  class  which  has  devoted 
a  little  time  to  graphs,  is  to  consider  the  graphic  sig- 
nificance of  each  new  equation  obtained  in  the  solu- 
tion of  a  pair  of  simultaneous  equations  involving  two 
unknowns.  Each  equation  properly  derived  must  rep- 
resent a  graph  passing  through  the  intersections  of 
the  graphs  corresponding  to  the  first  two.  E.g.  : 

1.  Given  X?  +y*  =  9, 

2.  and  x  +  y  =  3. 

1  Chrystal,  Presidential  address,  1885. 


TYPICAL  PARTS  OF  ALGEBRA  211 

3.  Then  x*  —  xy  +y*  =  3,  by  division. 

4.  x*  +  2xy  +y*  =  9,  by  squaring  (2). 

5.  .'.  xy  =  2,  by  subtracting  and  dividing. 

Equation  (3)  represents  an  ellipse  which  passes 
through  all  the  intersections  of  the  graphs  (i)  and 
(2)  except  the  point  at  infinity;  (4)  represents  two 
parallel  lines,  only  one  of  which  passes  through  th*e 
intersections  of  (i)  and  (2);  (5)  is  an  hyperbola  pass- 
ing through  the  intersections  of  the  ellipse  (3)  and 
the  parallels  (4).  The  solution  then  passes  on  to  the 
intersection  of  the  straight  line  (2)  with  the  par- 
allels (x-yf  =  i.1 

In  general,  the  question  of  the  number  of  roots 
to  be  expected,  the  entrance  of  complex  roots  in 
pairs,  the  conditions  rendering  equations  simultane- 
ous, or  inconsistent,  or  impossible,  these  necessary 
and  not  particularly  difficult  bits  of  theory  are  made 
to  stand  out  much  more  clearly  by  the  use  of  the 
graph. 

Methods  of  elimination  —  Elementary  text-books  al- 
ways distinguish  several  cases  of  elimination  with 
respect  to  linear  equations.  These  are,  (i)  addition, 
(2)  subtraction,  (3)  comparison,  (4)  substitution,  and 
possibly  (5)  B^zout's  method.  If  those  who  love 
novelty  only  knew  it,  there  are  numerous  other 

1  A  problem  used  by  Professor  Beman  in  his  teachers'  course  in  algebra. 


212    THE  TEACHING  OF  ELEMENTARY  MATHEMATICS 

methods  which  might  be  brought  in  to  give  this  turn 
to  the  subject.1 

But  for  the  practical  purposes  of  a  beginner  there 
are  only  two  distinct  methods  of  much  value,  (i) 
that  of  addition,  under  which  subtraction  is  merely  a 
special  case,  because  the  sign  of  the  proper  multiplier 
to  be  employed  will  always  reduce  the  process  to 
one  of  addition ;  (2)  that  of  substitution,  under  which 
comparison  is  merely  a  special  case,  for  in  equating 
x—y  —  2  and  x=$y  +  4,  we  substitute  the  value  of 
x  just  as  much  as  we  compare  values.  Hence  in 
teaching  the  subject,  it  is  to  these  two  methods 
that  especial  attention  is  to  be  given,  the  other  plans 
suggested  by  the  text-book  being  shown  to  be  special 
cases.  Indeed,  before  the  pupil  leaves  the  subject  it 
might  not  be  going  too  far  to  show  that  the  method 
of  substitution  is  a  special  case  of  the  one  general 
method  of  addition.  V 

Of  equal  importance  with  the  existence  of  the  two 

methods  mentioned,  is   the   question   as   to   their  use. 

s\ 

The  pupil  will  easily  find  for  himself,  if  permitted  to 
do  so,  that  the  addition  method  is  usually  preferable, 
the  other  being  the  easier  only  in  special  cases,  as 
in  that  of  unit  coefficients,  or  in  finding  one  of  two 
values  after  the  other  has  been  ascertained. 

When  both  equations  are  of  the  second  degree,  the 
student  should  early  be  led  to  see  that  in  general  no 

1  See  Matthiessen,  for  example. 


TYPICAL  PARTS  OF  ALGEBRA  213 

solution  is  possible  by  quadratics,  arid  that  the  only 
cases  which  he  can  handle  with  any  certainty  are 
those  involving  homogeneous  or  symmetric  functions. 
The  methods  of  attacking  these  cases  are  well  known 
and  need  not  be  discussed  here.  But  in  the  case  of 
symmetric  equations  it  should  be  noted  that  most 
text-books  lose  sight  of  one  of  the  essential  features. 
By  the  very  nature  of  symmetry  the  roots  must  be 
identical.  Consider,  for  example,  the  set 


By  the  usual  method  x  is  found  to  be  £(—  19  ±  V329), 
5,  or  i,  four  results,  as  should  have  been  anticipated. 
It  therefore  follows,  without  substituting  or  applying 
any  special  devices,  that  y  has  identically  the  same 
values  because  of  the  symmetry  of  each  function  as 
to  x  and  y.  Of  course  the  particular  value  of  y  to 
be  taken  with  a  given  value  of  x  is  not  yet  deter- 
mined, but  this  is  usually  seen  at  once  by  looking 
at  the  two  equations.  The  failure  to  recognize  all 
this  results  in  serious  loss  of  time;  the  student  gets 
exercise,  it  is  true,  but  he  might  more  profitably  get 
it  by  solving  another  set  of  equations  than  by  failing 
to  appreciate  one  of  the  essential  parts  of  the  theory. 
Complex  numbers  —  As  already  stated,  it  is  only 
since  Gauss,  in  1832,  brought  before  the  mathemat- 
ical world  at  large  the  theory  which  Wessel  and 
Argand  had  developed,  that  the  complex  number  has 


214    THE  TEACHING  OF   ELEMENTARY   MATHEMATICS 

been  well  understood.  Even  now  it  is  only  slowly 
finding  its  way  into  elementary  text-books,  such  works 
usually  saying  (of  course  "between  the  lines"),  "Here 
is  V—  i,  and  we  do  not  know  what  it  means  or  what 
to  do  with  it,  and  we  will  hasten  over  it  with  as 
little  trouble  as  possible."  Where  in  the  course  in 
algebra  this  perfunctory  treatment  shall  be  given  has 
been  the  subject  of  not  a  little  discussion,  as  if  it 
made  any  difference.  If  the  student  is  to  receive 
nothing,  what  matter  whether  that  nothing  comes 
this  month  or  next? 

What,  then,  should  be  done  with  the  subject? 
When  should  it  be  introduced,  and  how  should  it  be 
explained  ? 

It  is  an  educational  maxim  already  several  times 
invoked  in  these  pages,  that  a  subject  is  best  intro- 
duced just  before  it  is  to  be  used.  As  soon  as  we 
reach  quadratic  equations  as  a  distinct  subject  we 
meet  complex  numbers.  Equations  like  xz  -f-  i  =  o, 
x*  4-  2  x  -{-  5  =  o,  and  in  general  x*  +  px  -f  q  =  o  where 
p2  <  4  q,  give  rise  to  roots  involving  imaginaries. 
Hence  it  follows  that  the  chapter  on  complex  num- 
bers logically  precedes  that  on  quadratic  equations. 
Whether  it  psychologically  precedes  depends  upon  its 
difficulty. 

The  difficulty  of  the  chapter  has  been  overrated 
because  it  is  only  recently  that  teachers  as  a  class 
have  known  anything  about  the  subject.  In  reality 


~>x 

or  IMF          \ 

UNIVERSITY  ) 
TYPICAL  PARTS  OF  ALGEBRA  215 

the  graphic  treatment  of  the  complex  number  is  no 
more  difficult  for  the  pupil  who  is  ready  to  begin 
quadratics  than  is  that  of  the  negative  number  to 
the  one  about  to  take  up  the  theory  of  subtraction. 
Teachers  are  therefore  urged,  even  at  the  expense  of 
a  week's  work  outside  the  text-book  —  if  that  be  a 
hardship  —  to  present  the  elements  of  this  graphic 
treatment.1 

The  applied  problems  of  algebra  are  usually 
even  more  objectionable  than  those  of  arithmetic. 
When  the  science  began  to  find  place  in  the  schools 
there  had  accumulated  a  large  number  of  examples 
which  by  arithmetic  were  puzzles,  but  by  algebra 
offered  little  difficulty.  These  were  incorporated  in 
the  new  science,  and  they  have  remained  there  by  the 
usual  influence  of  two  powerful  agents — the  conserv- 
atism of  teachers  and  the  various  kinds  of  state  ex- 
aminations. To  this  latter  influence  is  to  be  charged 
the  greatest  amount  of  blame  in  the  matter,  not  as 
to  the  individuals  who  set  the  examinations,  but  to 
the  inherent  evil  (possibly  a  necessary  one)  of  the 
system.  Certain  of  the  best  teachers  of  a  country 
know  that  time  is  wasted  over  some  particular  line 
of  problems ;  they  would  like  to  omit  them,  but  their 

1  One  of  the  best  elementary  presentations  of  the  subject  is  given  in 
Fine's  Number  System  of  Algebra,  Boston,  1890,  a  book  which  should  be 
upon  the  shelves  of  every  teacher  of  this  subject.  For  a  classroom  treat- 
ment, see  Beman  and  Smith's  Algebra,  Boston,  1900. 


2l6    THE  TEACHING  OF  ELEMENTARY  MATHEMATICS 

hands  are  tied  by  the  necessity  that  their  pupils  shall 
pass  a  certain  examination  (civil  service,  college  — 
for  these  are  often  among  the  most  objectionable, 
regents',  teachers',  etc.).  On  the  other  hand,  many 
of  the  examiners  are  among  the  most  progressive 
educators.  They  too  would  like  to  see  the  mathe- 
matical field  weeded  and  conservatively  sown  anew. 
But  their  hands  are  also  tied  by  the  system.  As  a 
progressive  English  examiner  once  remarked  to  the 
writer,  "I  know  that  this  problem  should  have  no 
place  in  the  examination,  but  I  cannot  replace  it  by 
a  modern  one  because  the  schools  are  not  up  to  such 
a  change;  their  text-books  do  not  prepare  for  it." 

Speaking  of  this  effect  of  the  examination,  Pro- 
fessor Chrystal  has  not  hesitated  to  express  his  views 
with  perfect  frankness.  "The  history  of  this  matter 
of  problems,  as  they  are  called,  illustrates  in  a  singu- 
larly instructive  way  the  weak  point  of  our  English 
system  of  education.  They  originated,  I  fancy,  in  the 
Cambridge  Mathematical  Tripos  Examination,  as  a 
reaction  against  the  abuses  of  cramming  book-work, 
and  they  have  spread  into  almost  every  branch  of 
science  teaching  —  witness  test-tubing  in  chemistry. 
At  first  they  may  have  been  a  good  thing;  at  all 
events  the  tradition  at  Cambridge  was  strong  in  my 
day  that  he  who  could  work  the  most  problems  in 
three  or  two  and  a  half  hours  was  the  ablest  man, 
and,  be  he  ever  so  ignorant  of  his  subject  in  its 


TYPICAL  PARTS  OF  ALGEBRA  217 

width  and  breadth,  could  afford  to  despise  those  less 
gifted  with  this  particular  kind  of  superficial  sharp- 
ness. But,  in  the  end,  it  all  came  to  the  same:  we 
prepared  for  problem-working  in  exactly  the  same 
way  as  for  book-work.  We  were  directed  to  work 
through  old  problem  papers,  and  study  the  style  and 
peculiarities  of  the  day  and  of  the  examiner.  The 
day  and  the  examiner  had,  in  truth,  much  to  do  with 
it,  and  fashion  reigned  in  problems  as  in  everything 
else."1  Still  more  pointedly  he  says:  "All  men 
practically  engaged  in  teaching,  who  have  learned 
enough,  in  spite  of  the  defects  of  their  own  early 
training,  to  enable  them  to  take  a  broad  view  of  the 
matter,  are  agreed  as  to  the  canker  which  turns 
everything  that  is  good  in  our  educational  practice 
to  evil.  It  is  the  absurd  prominence  of  written  com- 
petitive examinations  that  works  all  this  mischief. 
The  end  of  all  education  nowadays  is  to  fit  the 
pupil  to  be  examined;  the  end  of  every  examination 
not  to  be  an  educational  instrument,  but  to  be  an 
examination  which  a  creditable  number  of  men,  how- 
ever badly  taught,  shall  pass.  We  reap,  but  we  omit 
to  sow.  Consequently  our  examinations,  to  be  what 
is  called  fair  —  that  is,  beyond  criticism  in  the  news- 
papers—  must  contain  nothing  that  is  not  to  be  found 
in  the  most  miserable  text-book  that  any  one  can 
cite  bearing  on  the  subject.  .  .  .  The  result  of  all 

1  Presidential  address,  British  Association,  Section  A,  1885. 


218    THE  TEACHING  OF  ELEMENTARY  MATHEMATICS 

this  is  that  science,  in  the  hands  of  specialists,  soars 
higher  and  higher  into  the  light  of  day,  while  edu- 
cators and  the  educated  are  left  more  and  more  to 
wander  in  primeval  darkness." 1 

This  evil,  which  we  have  not  yet  the  ingenuity  to 
avoid,  stares  the  teacher  in  the  face  when  he  would 
replace  obsolete  matter  by  problems  which  have  the 
stamp  of  the  generation  in  which  we  live.  It  is  not 
that  these  problems  about  the  pipes  filling  the  cistern, 
the  hound  chasing  the  hare,  the  age  of  Demochares, 
and  the  number  of  nails  in  the  horse's  shoe,  are  not 
good  wit-sharpeners,  and  possessed  of  a  kind  of  in- 
terest; but  we  have  now  a  large  number  of  equally 
good  wit-sharpeners  possessed  of  a  living  interest, 
problems  relating  to  the  life  we  now  live,  and  to  the 
simple  science  the  pupil  is  now  studying.  "I  some- 
times feel  a  doubt,  however,"  says  a  recent  writer, 
"  whether  boys  really  enjoy  being  introduced  to  such 
exercises  as  'A  says  to  B,  how  much  money  have 
you  got  ? '  and  B  makes  a  very  singular  hypothetical 
reply;  or  to  the  fish  whose  body  is  half  as  long 
again  as  his  head  and  tail  together,  while  head  and 
tail  have  given  relations  of  magnitude.  I  cannot  but 
suspect  that  there  is  something  unpractical  in  these 
problems."2  These  historical  problems  have  some 
value  as  history  and  some  interest  from  their  very 

1  Presidential  address,  1885. 

3  Heppel,  G.,  in  the  Mathematical  Gazette,  February,  1895. 


TYPICAL  PARTS  OF  ALGEBRA  219 

absurdity,  but  it  is  to  be  hoped  that  the  rising  gene- 
ration of  teachers  may  see  them  laid  aside.  "  A  more 
rational  treatment  of  the  subject,  introducing  from 
the  beginning  reasoning  rather  than  calculation,  and 
applying  the  results  obtained  to  various  problems 
taken  from  all  parts  of  science,  as  well  as  from 
everyday  life,  would  be  more  interesting  to  the  stu- 
dent, give  him  really  useful  knowledge,  and  would  be 
at  the  same  time  of  true  educational  value." 1 

It  is  a  serious  question  whether  America,  following 
England's  lead,  has  not  gone  into  problem-solving 
altogether  too  extensively.  Certain  it  is  that  we  are 
producing  no  text-books  in  which  the  theory  is  pre- 
sented in  the  delightful  style  which  characterizes 
many  of  the  French  works  (for  example,  that  of 
Bourlet),  or  those  of  the  recent  Italian  school  (like 
Pincherle's  handbooks),  or,  indeed,  those  of  the  conti- 
nental writers  in  general.  "  In  short,  the  logic  of  the 
subject,  which,  both  educationally  and  scientifically 
speaking,  is  the  most  important  part  of  it,  is  wholly 
neglected.  The  whole  training  consists  in  example 
grinding.  What  should  have  been  merely  the  help 
to  attain  the  end  has  become  the  end  itself.  The 
result  is  that  algebra,  as  we  teach  it,  is  rules,  whose  ob- 
ject is  the  solution  of  examination  problems.  .  .  .  The 
result,  so  far  as  problems  worked  in  examinations  go, 
is,  after  all,  very  miserable,  as  the  reiterated  com- 

1  Ilenrici,  O.,  Presidential  address,  British  Association,  Section  A,  1883. 


22O    THE  TEACHING  OF  ELEMENTARY  MATHEMATICS 

plaints  of  examiners  show ;  the  effect  on  the  ex- 
aminee is  a  well-known  enervation  of  mind,  an 
almost  incurable  superficiality,  which  might  be  called 
Problematic  Paralysis  — a  disease  which  unfits  a  man 
to  follow  an  argument  extending  beyond  the  length 
of  a  printed  octavo  page.  .  .  .  Against  the  occa- 
sional working  and  propounding  of  problems  as  an 
aid  to  the  comprehension  of  a  subject,  and  to  the 
starting  of  a  new  idea,  no  one  objects,  and  it  has 
always  been  noted  as  a  praiseworthy  feature  of  Eng- 
lish methods,  but  the  abuse  to  which  it  has  run  is 
most  pernicious."1 

The  interpretation  of  solutions — Algebra  is  generous, 
says  D'Alembert;  it  often  gives  more  than  is  asked.2 
And  it  is  one  of  the  mysteries  which  teachers  and 
text-books  usually  draw  about  the  science,  that  some 
of  the  solutions  of  the  applied  problems  are  not 
usable,  are  meaningless. 

But  there  should  be  no  mystery  about  this.  It  is  a 
fact,  easily  explained,  that  it  is  not  at  all  difficult  to  put 
physical  limitations  on  a  problem  that  shall  render  the 
result  mathematically  correct  but  practically  impossible. 
For  example,  if  I  can  look  out  of  the  window  9  times  in 
2  seconds,  how  many  times  can  I  look  out  in  i  second, 
at  the  same  rate?  The  answer,  4^  times,  is  all  right 

1  Chrystal,  Presidential  address  of  1885. 

2  L'algebre    est  genereuse  j    elle   donne  souvent  plus    qu'on  ne    lui 
demande. 


TYPICAL  PARTS  OF  ALGEBRA  221 

mathematically,  but  physically  I  cannot  look  out  half  a 
time.  Similarly,  if  5  men  are  to  ride  in  2  carriages, 
how  many  will  go  in  each,  the  carriages  to  contain  the 
same  number  ?  Mathematically  the  solution  is  simple, 
but  a  physical  condition  has  been  imposed,  "the  car- 
riages to  contain  the  same  number,"  which  makes  the 
problem  practically  impossible.  A  few  such  absurd 
cases  take  away  all  the  mystery  attaching  to  results  of 
this  nature,  and  show  how  easy  it  is  to  impose  restric- 
tions that  exclude  some  or  all  results. 

For  example,  the  number  of  students  in  a  certain 
class  is  such  as  to  satisfy  the  equation  2;r2  — 33* 
—  140  =  o ;  how  many  are  there  ?  The  conditions  of 
the  problem  are  such  as  to  make  one  root,  20,  legiti- 
mate, but  the  other,  —  |,  meaningless.  Algebra 
has  been  generous;  it  has  given  more  than  was 
asked. 

Consider  also  the  problem,  A  father  is  53  years  old 
and  his  son  28 ;  after  how  many  years  will  the  father 
be  twice  as  old  as  the  son  ?  From  the  equation 
53  +  x  —  2(28  +  x)  we  have  x—  —  ^.  We  are  now 
under  the  necessity  of  either  (i)  interpreting  the  ap- 
parently meaningless  answer,  —  3  years  after  this  time, 
or  (2)  changing  the  statement  of  the  problem  to  avoid 
such  a  result.  Either  plan  is  feasible.  We  may  in- 
terpret "  —  3  years  after "  as  equivalent  to  "  3  years 
before,"  which  is  entirely  in  accord  with  the  notion 
of  negative  numbers;  or  we  may  change  the  problem 


222    THE  TEACHING  OF  ELEMENTARY  MATHEMATICS 

to  read,  "  How  many  years  ago  was  the  father  twice  as 
old  as  the  son."  Most  algebras  require  this  latter 
plan,  one  inherited  from  the  days  when  the  negative 
number  was  less  understood  than  now. 

"Unlike  other  sciences,  algebra  has  a  special  and 
characteristic  method  of  handling  impossibilities.  If 
this  problem  of  algebra  is  impossible,  if  that  equation 
is  insoluble,  instead  of  hesitating  and  passing  on  to 
some  other  question,  algebra  seizes  these  solutions  and 
enriches  its  province  by  them.  The  means  which  it 
employs  is  the  symbol." *  The  symbol  "  —  3,"  for  the 
number  of  years  after  the  present  time,  without  sense 
in  itself,  is  seized  and  turned  into  a  means  for  enriching 
the  domain  of  algebra  by  the  introduction  and  interpre- 
tation of  negative  numbers. 

The  further  interpretation  of  negative  results,  and 
the  discussion  of  the  results  of  problems  involving  lit- 
eral equations,  is  a  field  of  considerable  interest  and 
value;  but  since  most  text-books  furnish  a  sufficient 
treatment  of  the  subject,  it  need  not  be  considered 
here. 

Conclusion  —  The  few  topics  mentioned  in  this  chap- 
ter might  easily  be  extended.  It  would  be  suggestive 
to  dwell  upon  the  absurdity  of  drilling  a  pupil  upon  the 
two  artificially  distinct  chapters  on  surds  and  fractional 
exponents,  as  our  ancestors  used  to  separate  the  "rule 
of  three"  from  proportion  —  matters  explainable  only 
1  De  Campou. 


TYPICAL  PARTS  OF  ALGEBRA  223 

by  reviewing  their  history.  The  theory  of  fractions, 
the  common  fallacy  in  the  proof  of  the  binomial  the- 
orem for  general  exponents,  the  use  of  determinants, 
the  complete  explanation  of  division  or  involution,  the 
questions  of  zero,  of  infinity,  and  of  limiting  values  — 
these  and  various  other  topics  will  suggest  themselves 
as  worthy  a  place  in  a  chapter  of  this  kind.  But  the 
limitations  of  this  work  are  such  as  to  exclude  them. 
The  topics  already  discussed  are  types,  and  it  is  hoped 
that  they  may  lead  some  of  our  younger  teachers  of 
algebra  to  see  how  meagre  is  the  view  offered  by  many 
of  our  elementary  text-books,  how  much  interest  can 
easily  be  aroused  by  a  broader  treatment  of  the  simpler 
chapters,  and  how  necessary  it  is  to  guard  against  the 
dangers  of  the  slipshod  methods  and  narrow  views 
which  are  so  often  seen  in  the  schools.  As  algebra 
is  often  taught,  there  is  force  in  Lamartine's  accusa- 
tion, that  mathematical  teaching  makes  man  a  machine 
and  degrades  thought,1  and  there  is  point  to  the 
French  epigram,  "  One  mathematician  more,  one  man 
less."2 

1  L'enseignement  mathematique  fait  1'homme  machine  et  degrade  la 
pensee.  Rebiere's  Mathematiques  et  mathematicians,  p.  217. 

8  Un  mathematician  de  plus,  un  homme  de  inoins.  Dupanloup.  Quoted 
in  Rebiere,  ib.,  p.  217. 


CHAPTER   IX 
THE  GROWTH  OF  GEOMETRY 

Its  historical  position  —  Roughly  dividing  elementary 
mathematics  into  the  science  of  number,  the  science 
of  form,  and  the  science  of  functions,  the  subject  has 
developed  historically  in  this  order.  Hence  the  chrono- 
logical sequence  would  lead  to  the  consideration  of 
geometry  before  algebra,  not  only  in  the  curriculum, 
but  in  a  work  of  this  nature.  The  somewhat  closer 
relation  of  arithmetic  and  algebra,  however,  explains 
the  order  here  followed,  if  explanation  is  necessary 
for  a  matter  of  so  little  moment. 

Reserving  for  the  following  chapter,  as  was  done 
with  algebra,  the  question  of  the  definition  of  geom- 
etry, we  may  consider  by  what  steps  the  science  as- 
sumed its  present  form.  We  shall  thus  understand 
more  clearly  the  limitations  which  the  definition  will  be 
seen  to  place  upon  the  subject,  we  shall  see  the  trend 
which  the  science  is  taking,  and  we  shall  the  more 
plainly  comprehend  the  nature  of  the  work  to  be 
undertaken  by  the  next  generation  of  teachers. 

The  dawn  of  geometry  —  The  world  has  always 
tended  to  deify  the  mysterious.  The  sun,  the  stars, 
fire,  the  sea,  life,  death,  number  —  these  have  all 

224 


THE  GROWTH  OF  GEOMETRY  22$ 

played  parts  in  the  great  religious  drama.  Whether  it 
be  that  the  plains  of  Babylon  were  especially  adapted 
to  the  care  of  flocks,  or  that  the  purity  of  the  Egyp- 
tian atmosphere  led  to  the  study  of  the  heavenly 
bodies,  or  that  both  of  these  causes  played  their  parts, 
certain  it  is  that  in  Mesopotamia  and  along  the  Nile 
a  primitive  astronomy  developed  at  an  early  period  and 
took  its  place  as  a  part  of  the  store  of  ancient  reli- 
gious mysteries.  With  it  went  some  rude  knowledge  of 
geometry,  the  demands  of  practical  life  also  creating  from 
time  to  time  an  empirical  science  of  simple  mensuration. 
Thus  among  the  Babylonians  we  find  the  circle  of 
the  year  early  computed  at  360  days  (whence  the  circle 
was  divided  into  that  number  of  degrees),  and  later, 
as  astronomical  observation  improved,  at  more  nearly 
the  correct  number.1  The  Babylonian  monuments  so 
often  picture  chariot  wheels  as  divided  into  sixths,  that 
it  is  probable  that  the  method  of  dividing  the  circum- 
ference into  sixths  by  means  of  striking  circles  was 
early  known,  a  method  which  carries  with  it  the  inscrip- 
tion of  the  regular  hexagon.  This  would  show  that  the 
circumference  is  a  little  more  than  6r  or  3^,  but  TT 
seems  generally  to  have  been  taken  as  3  by  them  and 
their  neighbors.2 

1  Hankel,  Zur  Geschichte  der  Mathematik,  p.  71,  for  the  pre-scientific 
geometry. 

2  I    Kings  vii,  23;    2  Chron.   iv,  2.     "What   is  three   handbreadths 
around  is  one  handbreadth  through."    Talmud. 

Q 


226    THE  TEACHING  OF  ELEMENTARY  MATHEMATICS 

The  Egyptians  were  particular  as  to  the  proper 
orientation  of  their  temples,  a  custom  still  considered  of 
moment  by  a  large  part  of  the  religious  world.  The 
meridian  line  was  established  by  the  pole  star,  and  for 
the  east  and  west  line  the  temple  builders  were  early 
aware  of  a  rule  still  used  by  surveyors  in  laying  off  a 
perpendicular.  The  present  plan  is  to  take  eight  links 
of  a  surveyor's  chain,  place  the  ends  of  the  chain  four 
links  apart,  and  stretch  it  with  a  pin  at  the  fifth  link, 
this  forms  a  right-angled  triangle  with  sides  3,  4,  5.  The 
Egyptians  did  the  same  in  building  their  temples,  and 
the  harpedonaptae  ot  "rope-stretchers"  laid  out  the  plans, 
as  a  civil  engineer  lays  out  those  for  a  building  to-day.1 

The  scholars  of  the  Nile  valley  also  possessed  some 
knowledge  of  the  rudiments  of  trigonometry,2  and 
their  approximation  to  the  value  of  TT  was  not  im- 
proved for  many  centuries.  Ahmes  gave  the  value 
TT=  (-1^-)2=  3.1605,  a  remarkably  good  approximation 
for  a  period  when  geometry  was  little  more  than  men- 
suration. He  was  not  so  fortunate  in  all  of  his  rules, 
for  example  in  the  one  for  finding  the  area  of  an 
isosceles  triangle,  which  required  the  multiplication 
of  the  measure  of  half  the  base  by  that  of  one  of 
the  equal  sides. 

1  This  interpretation  of  the  Greek  harpedonaptae  is  one  of  Professor 
Cantor's  ingenious  discoveries.    Cantor,  I,  p.  62. 

2  A  brief  summary  is  given  in  Gow,  History  of  Greek  Mathematics, 
p.  128. 


THE  GROWTH  OF  GEOMETRY  227 

The  indebtedness  of  the  Greeks,  who  were  the 
next  to  take  up  geometry,  to  the  Egyptians  is  well 
summarized  by  Gow :  "  It  remains  only  to  cite  the 
universal  testimony  of  Greek  writers,  that  Greek  geom- 
etry was,  in  the  first  instance,  derived  from  Egypt, 
and  that  the  latter  country  remained  for  many  years 
afterward  the  chief  source  of  mathematical  teaching. 
The  statement  of  Herodotus  on  this  subject  has 
already  been  cited.  So  also  in  Plato's  '  Phaedrus,' 
Socrates  is  made  to  say  that  the  Egyptian  god  Theuth 
first  invented  arithmetic  and  geometry  and  astronomy. 
Aristotle  also  ('Metaphysics,'  I,  i)  admits  that  geom- 
etry was  originally  invented  in  Egypt;  and  Eudemus 
expressly  declares  that  Thales  studied  there.  Much 
later  Diodorus  (70  B.C.)  reports  an  Egyptian  tradition 
that  geometry  and  astronomy  were  the  inventions  of 
Egypt,  and  says  that  the  Egyptian  priests  claimed 
Solon,  Pythagoras,  Plato,  Democritus,  CEnopides  of 
Chios,  and  Eudoxus  as  their  pupils.  Strabo  gives 
further  details  about  the  visits  of  Plato  and  Eudoxus. 
.  .  .  Beyond  question,  Egyptian  geometry,  such  as 
it  was,  was  eagerly  studied  by  the  early  Greek  phi- 
losophers, and  was  the  germ  from  which  in  their  hands 
grew  that  magnificent  science  to  which  every  English- 
man is  indebted  for  his  first  lessons  in  right  seeing  and 
thinking." l 

The  Greeks    were,  however,   the    first    to    create  a 

1  History  of  Greek  Mathematics,  p.  131. 


228    THE  TEACHING  OF  ELEMENTARY   MATHEMATICS 

science  of  geometry.  Thales  (  —  640,  —  548),  having 
through  trade  secured  the  financial  means  for  study, 
travelled  in  Egypt  for  the  purpose  of  acquiring  the 
mathematical  lore  of  the  priests,  giving  quite  as  much 
as  he  received,  and  finally  established  a  school  in 
Asia  Minor,  where  the  first  important  scientific  in- 
vestigations in  geometry  were  made. 

The  most  noted  pupil  of  Thales  was  Pythagoras, 
who  was  with  him  for  a  short  time  at  least  and  who 
was  advised  by  him  to  continue  his  studies  in  Egypt. 
The  school  which  Pythagoras  afterward  opened  in 
Croton,  in  Southern  Italy,  was  one  of  the  most 
famous  of  all  antiquity,  and  here  geometry  was  seri- 
ously cultivated.  Here  were  proved  the  following 
propositions,  among  others :  the  plane  about  a  point 
is  filled  by  six  equilateral  triangles,  four  squares  or 
three  regular  hexagons ;  the  sum  of  the  interior 
angles  of  a  triangle  is  two  right  angles;  the  sum  of 
the  squares  on  the  sides  of  a  right-angled  triangle 
equals  the  square  on  the  hypotenuse,  a  fact  known  to 
the  Egyptians  but  first  proved  by  the  Pythagoreans. 

From  now  on  until  the  third  century  before  Christ 
Greek  geometry  passed  through  its  golden  age. 1 


1  For  detailed  notes  as  to  the  discoveries  of  the  Greeks  see  Allman, 
G.  J.,  Greek  Geometry  from  Thales  to  Euclid;  Bretschneider,  Die 
Geometric  und  die  Geometer  vor  Eukleides,  Leipzig,  1870;  Gow,  J., 
History  of  Greek  Mathematics,  Cambridge,  1884;  Beman  and  Smith's 
translation  of  Fink's  History  of  Mathematics,  Chicago,  1900 ;  Chasles, 


THE  GROWTH  OF  GEOMETRY  229 

The  principal  discoveries  in  elementary  geometry 
were  made  in  the  two  centuries  from  —  500  to  —  300, 
and  were  crystallized  in  logical  form  by  Euclid,  who 
taught  in  the  famous  school  at  Alexandria  about 
—  300.  During  this  period,  owing  to  the  vast  extent 
of  the  field  opened  up  by  the  study  of  conic  sections, 
Plato  (—429,  —348)  placed  a  definite  limit  upon  elemen- 
tary geometry,  allowing  only  the  compasses  and  the 
unmarked  straight-edge  as  instruments  for  the  con- 
struction of  figures. 

So  complete  as  a  specimen  of  logic  was  Euclid's 
treatment  of  elementary  geometry,  that  it  has  been 
used  as  a  text-book,  with  slight  modifications,  for 
over  two  thousand  years.  This  use  has  not,  however, 
been  general.  Indeed,  it  has  needed  the  exertions  of 
men  like  Hoiiel  in  France  and  Loria1  in  Italy,  and 
other  Continental  writers,  to  recall  from  time  to  time 
the  merits  of  Euclid  to  the  educational  world.  But 
in  England  Euclid  still  holds  a  sway  that  is  prac- 
tically absolute.2 

The  influence  of  the  Greek  writers  is  still  seen  in  the 

M.,  Apercu  historique  sur  1'origine  .  .  .  de  Geometric,  Paris,  2.  ed.,  1875; 
and  of  course  Cantor  and  Ilankel. 

1  Delia  varia  fortuna  di  Euclide  in  relazione  con  i  problemi  dell'  In- 
segnamento  Geometric©  Elementare,  Rome,  1893. 

2  Teachers  who  care  to  enter  into  the  merits  of  the  controversy  over 
Euclid  may  make  a   pleasant   beginning,  and  at   the   same   time   may 
see  the  mean   between   Dodgson   the   mathematician   and   Carroll   the 
writer  of  children's  stories  (as  Alice  in  Wonderland)  by  reading  Dodg- 
son, C.  L.,  Euclid  and  his  Modern  Rivals,  London. 


230    THE  TEACHING  OF  ELEMENTARY  MATHEMATICS  . 

nomenclature  of  the  science  the  world  over.  Because 
the  ancients  had  no  printing,  and  found  it  convenient 
to  have  the  rolls,  which  made  their  volumes,  somewhat 
brief,  the  word  "book"  came  to  apply  to  part  of  a 
treatise.  Thus  we  have  the  books  of  the  "  ^Eneid,"  of 
the  "  Iliad,"  and  of  treatises  on  geometry,  astronomy, 
etc.  The  word  has  been  preserved  in  the  divisions  of 
most  elementary  geometries  as  a  matter  of  interesting 
history.  Thus  Euclid's  first  book  is  chiefly  upon 
straight  lines  and  the  congruence  of  rectilinear  figures ; 
the  second  is  devoted  to  the  next  subject  of  which  the 
student  has  already  some  knowledge  —  rectangles  ;  the 
third  to  circles,  and  so  on.  With  doubtful  judgment 
some  of  our  modern  writers  have  followed  Legendre  in 
reversing  the  order  in  the  second  and  third  books, 
placing  circles  before  rectangles,  the  less  known  and 
more  difficult  concept  before  the  more  familiar  and 
simple. 

Many  other  words,  unlike  "book,"  are  distinctly 
Greek,  as,  for  example,  "theorem,"  "axiom,"  "scho- 
lium "  (happily  going  out  of  fashion),  "  trapezoid," 
"parallelogram,"  "parallelepiped"  (often  given  the 
unscientific  spelling  " parallelepiped  "),  "hypotenuse" 
(still  occasionally  spelled  with  an  h,  though  unscien- 
tifically so),  etc.  In  many  cases,  however,  the  Latin 
forms  have  displaced  the  Greek,  as  in  "  triangle " 
(rather  more  Latin  than  Greek),  "quadrilateral," 
"base,"  "circumference,"  "vertex,"  "surface,"  etc. 


THE  GROWTH  OF  GEOMETRY  231 

After  the  death  of  Archimedes  (  —  212),  to  whom  we 
owe  the  first  fruitful  scientific  attempts  at  the  mensura- 
tion of  the  circle,  geometry  seems  to  have  exhausted 
itself.  Excepting  a  few  sporadic  discoveries,  it  remained 
stagnant  for  nearly  two  thousand  years.  It  was  not 
until  the  seventeenth  century  that  any  great  advance 
was  made,  a  century  which  saw  the  discovery  of  analytic 
geometry  at  the  hands  of  Descartes,  the  revival  of  pure 
geometry  through  the  labors  of  Pascal  and  his  contem- 
poraries, and  which  saw  but  failed  to  recognize  the 
foundation  of  projective  geometry  in  the  works  of 
Desargues. 

Recent  geometry  —  The  nineteenth  century  has  seen 
a  notable  increase  of  interest  in  the  geometry  of  the 
circle  and  straight-edge,  a  geometry  which  can,  how- 
ever, hardly  be  called  elementary  in  the  ordinary  sense. 
France  has  been  the  leader  in  this  phase  of  the  subject, 
with  England  and  Germany  following.  Carrying  out 
the  suggestion  made  by  Desargues  in  the  seventeenth 
century,  Chasles,  about  the  middle  of  the  nineteenth  cen- 
tury, developed  the  theory  of  anharmonic  ratio,  making 
this  the  basis  of  what  may  be  designated  modern  geom- 
etry. Brocard,  Lemoine,  and  Neuberg  have  been  largely 
instrumental  in  creating  a  geometry  of  the  circle  and 
the  triangle,  with  special  reference  to  certain  interesting 
angles  and  points.  How  much  of  all  this  will  find  its 
way  into  the  elementary  text-books  of  the  next  genera- 
tion, replacing,  as  it  might  safely  do,  some  of  the  work 


232    THE  TEACHING  OF  ELEMENTARY  MATHEMATICS 

which  we  now  give,  it  is  impossible  to  say.  The  teacher 
who  wishes  to  become  familiar  with  the  elements  of 
this  modern  advance  could  hardly  do  better  than  read 
Casey's  Sequel  to  Euclid.1 

Along  more  advanced  lines  the  progress  of  geometry 
has  been  very  rapid.  The  labors  of  Mobius,  Pliicker, 
Steiner,  and  Von  Staudt,  in  Germany,  have  led  to  regions 
undreamed  of  by  the  ancients.  This  work  is  not,  how- 
ever, in  the  line  of  elementary  geometry,  and  therefore 
has  no  place  in  the  present  discussion.2 

Among  the  improvements  which  affect  the  teaching 
of  the  elementary  geometry  of  to-day,  a  few  deserve 
brief  mention.  Among  these  is  the  contribution  of 
"Mobius  on  the  opposite  senses  of  lines,  angles,  sur- 
faces, and  solids;  the  principle  of  duality  as  given  by 
Gergonne  and  Poncelet;  the  contributions  of  De  Mor- 
gan to  the  logic  of  the  subject;  the  theory  of  trans- 
versals as  worked  out  by  Monge,  Brianchon,  Servois, 
Carnot,  Chasles,  and  others;  the  theory  of  the  radical 
axis,  a  property  discovered  by  the  Arabs,  but  intro- 
duced as  a  definite  concept  by  Gaultier  (1813)  and 
used  by  Steiner  under  the  name  of  'line  of  equal 
power';  the  researches  of  Gauss  concerning  inscrip- 
tible  polygons,  adding  the  17-  and  25/-gon  to  the  list 
below  the  looo-gon;  .  .  .  and  the  researches  of  Muir 
on  stellar  polygons.  .  .  . 

1  London,  fifth  edition,  1888. 

2  For  a  brief  review  of  the  subject,  see  the  author's  article  in  Merriman 
and  Woodward's  Higher  Mathematics,  New  York,  1896,  p.  558. 


THE  GROWTH  OF  GEOMETRY  233 

"In  recent  years  the  ancient  problems  of  trisecting 
an  angle,  doubling  the  cube,  and  squaring  the  circle 
have  all  been  settled  by  the  proof  of  their  insolubility 
through  the  use  of  compasses  and  straight-edge." l 

Non-Euclidean  geometry  —  "The  non-Euclidean  ge- 
ometry is  a  natural  result  of  the  futile  attempts  which 
had  been  made  from  the  time  of  Proklos  to  the  opening 
of  the  nineteenth  century  to  prove  the  fifth  postulate 
(also  called  the  twelfth  axiom,  and  sometimes  the  elev- 
enth or  thirteenth)  of  Euclid."  This  is  essentially  the 
postulate  that  through  a  point  one  and  only  one  line  can 
be  drawn  parallel  to  a  given  line.  The  first  scientific 
investigation  of  this  part  of  the  foundation  of  geometry 
was  made  by  Saccheri  (1733).  The  matter  was  brought 
to  its  final  stage  by  Lobachevsky  and  Bolyai  about  1825, 
and  the  result  is  a  perfectly  consistent  geometry  denying 
the  validity,  or  the  necessity,  of  the  postulate  in  ques- 
tion.2 

1  Smith,  D.  E.,  History  of  Modern   Mathematics,   in  Merriman  and 
Woodward's  work  cited,  p.  564.     On  the  impossibility  of  solving  the  prob- 
lems mentioned,  see  Beman  and  Smith's  translation  of  Klein's  Famous 
Problems  of  Elementary  Geometry,  Boston,  1896. 

2  Smith,  D.  E.,  History  of  Modern  Mathematics,  p.  565. 


CHAPTER  X 

WHAT  is  GEOMETRY  ?     GENERAL  SUGGESTIONS  FOR 
TEACHING 

Geometry  defined  —  The  etymology  of  a  word  is  often 
far  from  giving  its  present  meaning.  We  have  already 
seen  this  in  the  case  of  "algebra"  and  "algorism" 
(p.  151).  Geometry  means  earth-measure  (77),  the  earth, 
+  fjierpelv,  to  measure),  and  probably  took  this  name  be- 
cause, in  its  prescientific  stage,  it  was  what  we  would 
now  call  by  the  unexpressive  term  "  surveying."  It 
came  to  mean,  among  the  Greeks,  the  science  of  fig- 
ures or  of  extent,  and  this  general  idea  still  obtains. 

More  specifically  we  may  say :  "  By  the  observation 
of  objects  about  us  we  arrive  at  the  concept  of  the  space 
in  which  we  live  and  in  which  these  objects  have  a  cer- 
tain extent.  We  are  aware  at  the  same  time  that  they 
have  a  form.  These  forms  are  infinitely  varied,  but 
certain  of  them  strike  us  by  their  regularity." 1  This 
regularity  is  rather  apparent  than  real,  and  the  appear- 
ance leads  us  to  make  certain  abstractions,  as  of  straight 
line,  circle,  square,  etc.,  forms  not  met  in  nature.  "  Just 
as  the  abstractions  made  concerning  collections  of 
objects2  are  the  basis  of  our  arithmetical  ideas,  so  the 

1  Laisant,  p.  89.  2  See  p.  100. 

234 


WHAT  IS  GEOMETRY  235 

abstractions  made  concerning  forms  are  the  origin  of 
our  conceptions  of  geometry." 1  Hence  the  science  of 
geometry  is  the  science  of  certain  abstractions  which  the 
mind  makes  concerning  form.  As  Laplace  says :  "  In 
order  to  understand  the  properties  of  bodies,  we  have 
first  to  cast  aside  their  particular  properties  and  to  see 
in  them  only  an  extended  figure,  movable  and  impene- 
trable. We  must  then  ignore  these  last  two  general 
properties  and  consider  the  extent  only  as  a  figure.  The 
numerous  relations  presented  under  this  last  point  of 
view  form  the  object  of  geometry."  * 

Elementary  geometry,  however,  limits  itself  to  com- 
paratively few  of  these  forms.  As  already  stated,  the 
great  field  opened  by  the  study  of  conies  and  higher 
plane  curves  led  Plato  to  limit  elementary  plane 
geometry  to  those  figures  which  can  be  constructed  by  \ 
the  use  of  the  compasses  and  the  unmarked  straight-  \ 
edge.  As  solid  geometry  has  gradually  developed,  it 
has  been  looked  upon  as  limited  to  figures  analogous 
to  those  of  plane  geometry,  the  sphere  analogous  to 
the  circle,  the  plane  to  the  straight  line,  etc.,  with 
the  addition  of  the  prism,  pyramid,  cone,  and  cylin- 
der. Euclid,  caring  little  for  the  practical  demands 
of  mensuration,  paid  almost  no  attention  to  solid 
geometry;  but  the  subject  has  assumed  much  prom- 
inence in  the  nineteenth  century,  without,  however, 

1  Laisant,  p.  89. 

*Dauge,  F.f  Methodologie,  p.  161. 


236    THE  TEACHING  OF  ELEMENTARY  MATHEMATICS 

having  its  limits  clearly  defined.  For  example,  whether 
a  cone  with  a  non-circular  directrix  shall  be  admitted 
is  an  unsettled  question ;  for  purposes  of  simple  men- 
suration of  volume  it  might  deserve  a  place,  but  hardly 
so  unless  the  mensuration  of  a  non-circular  curvilinear 
plane  figure  (its  base)  is  also  admitted. 

Limits  of  plane  geometry  —  But  elementary  geom- 
etry is  not  only  quite  uncertain  with  respect  to  the 
extent  of  the  portion  devoted  to  solids ;  the  recent 
additions  to  plane  geometry,  referred  to  in  Chapter 
IX,  have  made  the  limits  of  that  portion  of  the 
science,  as  to  its  "elements,"  even  more  undefined. 
With  the  recent  "geometry  of  the  triangle,"  as  it 
is  sometimes  called,  the  extent  of  the  subject  is  far 
beyond  the  possibilities  of  the  secondary  curricu- 
lum. It  cannot  all  be  excluded,  for  we  have  long 
since  introduced  the  notions  of  orthocentre,  centroid, 
ex-centre,  etc.,  but  just  what  shall  be  admitted  by 
the  next  generation  is  quite  uncertain,  as  would  be 
expected  in  view  of  the  fact  that  the  development 
is  so  recent.  Suffice  it  to  say  that  at  present  there  is 
no  general  agreement  as  to  what  constitutes  element- 
ary geometry,  save  this  —  that  it  shall  cover  substan- 
tially the  ground  of  Euclid's  "  Elements,"  plus  a  little 
work  on  loci,  the  mensuration  of  the  circle,  and  the  men- 
suration of  certain  common  solids.  From  this  state- 
ment, the  futility  of  attempting  a  scientific  definition  of 
the  elementary  geometry  of  the  schools  is  apparent. 


WHAT  IS  GEOMETRY  237 

The  reasons  for  studying  geometry,  as  for  studying 
arithmetic,  are  twofold.  We  have  the  practical  side 
of  the  subject  in  simple  mensuration,  and  we  have 
the  culture  side  in  the  logic  which  enters  into  it  to 
such  a  marked  degree. 

The  most  practical  part  of  mensuration  is  usually 
taught  in  connection  with  arithmetic,  formerly  by 
mere  rule,  now  with  the  models  in  hand  and  with  a 
semi-scientific  deduction  of  a  few  necessary  formulae. 
To  drop  the  science  there,  would  be  to  lose  its  chief 
value,  to  do  what  the  English  schools  do  with  solid 
geometry  —  a  mistake  also  often  made  in  our  Eastern 
states,  though  not  in  the  West.  The  danger  of  doing 
nothing  with  solid  geometry  save  in  the  way  of  men- 
suration, is  suggested  by  Professor  Henrici  in  these 
words  (referring  to  the  English  schools):  "Most  of 
all,  perhaps,  solid  geometry  has  suffered,  because 
Euclid's  treatment  of  it  is  scanty,  and  it  seems 
almost  incredible  that  a  great  part  of  it — the  men- 
suration of  areas  of  simple  curved  surfaces  and  of 
volumes  of  solids  —  is  not  included  in  ordinary  school 
teaching.  The  subject  is,  possibly,  mentioned  in 
arithmetic,  where,  under  the  name  of  mensuration,  a 
number  of  rules  are  given.  But  the  justification  of 
these  rules  is  not  supplied,  except  to  the  student 
who  reaches  the  application  of  the  integral  calculus; 
and  what  is  almost  worse  is  that  the  general  relation 
of  points,  lines,  and  planes,  in  space,  is  scarcely 


238    THE  TEACHING  OF   ELEMENTARY  MATHEMATICS 

touched  upon,  instead  of  being  fully  impressed  on 
the  student's  mind."  l 

The  culture  value,  which  is  almost  the  only  one 
which  formal,  demonstrative  geometry  has,  includes  two 
phases.  In  the  first  place,  we  need  to  know  geometry 
for  general  information,  because  the  rest  of  the  world 
knows  something  of  it.  It  must  be  admitted,  however, 
that  this  is  not  a  very  determining  reason,  for  it  is  one 
which  would  justify  keeping  any  traditional  subject  in 
the  curriculum. 

The  second  and  vitally  important  culture  phase  is 
that  of  the  logic  of  geometry.  Before  Euclid,  probably 
most  of  his  propositions  were  known ;  but  it  was  he 
who  arranged  them  in  the  order  and  with  the  demon- 
strations which  have  made  his  work  one  of  the  most 
admired  specimens  of  logic  that  have  ever  been  pro- 
duced. And  this  logic  has  given  added  significance 
and  beauty  to  the  truths  themselves.  "  They  enrich  us 
by  our  mere  contemplation  of  them.  In  this  connection 
I  wish  to  quote  the  beautiful  poem  'Archimedes  and 
the  Student,'  by  Schiller : 

"  To  Archimedes  once  came  a  youth,  who  for  knowledge  was  thirst- 
ing* 

Saying,  i  Initiate  me  into  the  science  divine, 

Which  for  my  country  has  borne  forth  fruit  of  such  wonderful 
value, 

And  which  the  walls  of  the  town  'gainst  the  Sambuco  protects.' 

1  Presidential  address,  1883. 


WHAT  IS  GEOMETRY  239 

'  Call'st  thou  the  science  divine  ?  It  is  so,1  the  wise  man  responded ; 

*  But  it  was  so,  my  son,  ere  it  availed  for  the  town. 

Would'st  thou  have  fruit  from  her,  only?  even  mortals  with   that 

provide  thee ; 
Would'st  thou  the  goddess  obtain  ?  seek  not  the  woman  in  her  ! ' " l 

ji 
Here,  then,  is  the  dominating  value  of  geometry,  its 

value  as  an  exercise  in  logic,  as  a  means  of  mental 
training,  "as  a  discipline  in  the  habits  of  neatness, 
order,  diligence,  and,  above  all,  of  honesty.  The 
fact  that  a  piece  of  mathematical  work  must  be  definitely 
right  or  wrong,  and  that  if  it  is  wrong  the  mistake  can 
be  discovered,  may  be  made  a  very  effective  means  of 
conveying  a  moral  lesson."2  Without  this  aim  well 
fixed  in  mind,  the  teacher  is  like  a  mariner  without  a 
compass ;  he  knows  not  whither  he  is  to  go ;  or,  if  he 
has  some  confused  idea  of  the  haven,  he  has  not  the 
means  to  find  his  way  thither. 

Having  now  considered  the  nature  of  elementary 
geometry,  and  the  reasons  for  teaching  the  science, 
the  question  arises  as  to  the  general  method  of  pre- 
senting it. 

Geometry  in  the  lower  grades  —  While  educators  differ 
materially  as  to  the  method  of  presenting  the  subject  of 
demonstrative  geometry,  this  being  still  an  open  question 
for  the  coming  generation  to  consider,  it  is  generally 

1  Schwatt,  I.  J.,  Some  Considerations  showing  the  Importance  of  Mathe- 
matical Study,  Philadelphia,  1895. 

8  Mathews,  G.  B.,  in  The  School  World,  Vol.  I,  p.  129  (April,  1899). 


240    THE  TEACHING  OF  ELEMENTARY   MATHEMATICS 

agreed  that  some  of  the  elementary  concepts  of  the 
science  should  be  acquired  in  the  lower  grades.  This 
view  was  long  ago  held  by  Rousseau.  "  I  have  said," 
he  remarks,  "  that  geometry  is  not  adapted  to  children ; 
but  this  is  our  fault.  We  seem  not  to  comprehend  that 
their  method  is  not  ours,  and  that  what  should  be  for  us 
the  art  of  reasoning  should  be  for  them  merely  the  art 
of  seeing.  Instead  of  thrusting  our  method  upon  them, 
we  would  do  better  to  adopt  theirs.  .  .  .  For  my  pupils, 
geometry  is  merely  the  art  of  handling  the  rule  and 
compasses." 1  Lacroix,  one  of  the  best  teachers  of 
mathematics  at  the  opening  of  the  nineteenth  century,2 
recognized  the  same  principle  when  he  said  :  "  Geometry 
is  possibly  of  all  the  branches  of  mathematics  that  which 
should  be  understood  first.  It  seems  to  me  a  subject 
well  adapted  to  interest  children,  provided  it  is  presented 
to  them  chiefly  with  respect  to  its  applications.  .  .  .  The 
operations  of  drawing  and  of  measuring  cannot  fail  to 
be  pleasant,  leading  them,  as  by  the  hand,  to  the  science 
of  reasoning."  Such  was  also  the  scheme  laid  out  by 
the  mathematician  Clairaut  and  approved  by  Voltaire, 
but  in  practice  it  has  not  been  systematically  followed 
by  the  teaching  profession. 

Laisant,  whose   rank   as   a    mathematician   and   an 

1  Rebiere,  A.,  Mathematiques  et  mathematicians,  p.  103. 

2  His  Essais  sur  1'enseignement  en  general,  et  sur  celui  des  mathe- 
matiques  en  particulier,  Paris,  1805,  was  one  of  the  earliest  works  of  any 
value  on  the  teaching  of  mathematics. 


WHAT  IS  GEOMETRY  24! 

educator  justifies  the  frequent  reference  to  his  name, 
thus  expresses  his  views:  "The  first  notions  of  ge- 
ometry should  be  given  to  the  child  along  with  the 
first  notions  of  algebra,  following  closely  upon  the 
beginning  of  theoretical  arithmetic  (rarithmetique 
raisonn^e).  But  just  as  there  must  be  a  preliminary 
preparation  for  arithmetic,  namely  practical  calcula- 
tion, so  theoretical  geometry  should  be  preceded  by 
the  practice  of  drawing.  The  habit  acquired  in 
childhood,  of  drawing  figures  neatly  and  with  sen- 
sible exactness,  would  be  of  great  assistance  later  in 
the  development  of  the  various  chapters  of  geometry. 
The  one  who  defined  geometry  as  the  art  of  correct 
reasoning  on  bad  figures,  was  altogether  wrong.  We 
never  reason  save  on  abstractions,  and  figures  are 
never  exact;  but  when  the  inaccuracy  is  too  manifest, 
when  the  drawings  are  poorly  executed  and  appear 
confused,  this  confusion  of  form  readily  leads  to  that 
of  reasoning  and  tends  to  conceal  the  truth.  Indeed 
there  are  cases  where  a  poorly  drawn  figure  leads 
by  logical  reasoning  to  manifest  absurdities.1  The 
first  education  in  geometry  should  therefore  be  under- 
taken, as  in  the  case  of  practical  computation,  with 
the  child  who  knows  how  to  read  and  write  the 
language  —  that  is,  who  knows  drawing.  .  .  .  Advan- 
tage should  be  taken  in  this  drawing  of  figures,  to 

1  Two  interesting  illustrations  of  this  fact  are  given  in  Ball's  Mathe- 
matical Recreations,  London,  1892,  p.  32. 

R 


242    THE  TEACHING  OF  ELEMENTARY  MATHEMATICS 

/  give  to  the  child  the  nomenclature  of  a  large  number 
of  geometric  concepts,  but  always  without  any  formal 
definitions." l 

The  views  of  Hoiiel,  one  of  the  best  teachers  of 
the  last  generation  in  France,  also  deserve  recogni- 
tion. "Let  us  imagine,"  he  says,  "the  possibility 
of  a  graduated  teaching  of  elementary  geometry 
carried  on  at  every  step  according  to  one  invariable 
plan,  always  governed  by  the  rules  of  severe  logic, 
but  with  the  difficulties  always  commensurate  with 
the  development  of  the  pupil's  mind.  For  such  a 
scheme  the  study  of  geometry  would  need  to  be  con- 
sidered from  various  points  of  view  corresponding  to 
the  various  degrees  of  initiation  of  the  pupil.  For 
beginners  it  would  be  necessary  first  of  all  to  famil- 
iarize them  with  the  various  geometric  figures  and 
their  names,  to  lead  them  to  know  facts  and  to 
understand  their  more  simple  and  immediate  appli- 
cations to  matters  of  daily  life.  We  ought  at  first 
to  multiply  the  axioms  and  to  employ,  in  place 
of  demonstrations,  experimental  truths,  analogy,  in- 
duction, always  remembering  that  this  method  of 
treatment  is  essentially  provisional.  .  .  .  The  first 
teaching  should  be  purely  experimental,  and  little  by 
little  the  pupil  should  come  to  see  that  all  truths 
need  not  necessarily  be  derived  from  experience,  but 
that  some  are  consequences  of  a  certain  number  of 

1  La  Mathematique,  p.  220. 


[UNIVERSITY   } 

V 

WHAT  IS  GEOMETRY  243 

others,  a  number  which  becomes  smaller  and  smaller 
as  one  advances  in  the  science  until  he  reaches  the 
fundamental  axioms."1 

The  ideas  above  set  forth  are  not  the  thoughts  of 
mere  theorizers;  they  have  been  carried  out  with 
more  or  less  success  in  many  European  and  Ameri- 
can schools.  The  outline  of  some  of  this  work  is 
given  in  the  subsequent  pages.  It  may,  however,  be 
said  for  the  lower  grades,  in  passing,  that  teachers 
should  insist  that  none  of  the  new  schemes  of  draw- 
ing which  apply  for  admission  to  the  schools  be 
lacking  in  this  particular.  The  study  of  the  common 
geometric  forms  in  the  early  years  is  too  valuable  to 
be  neglected. 

Intermediate  grades  —  The  next  step  in  the  work 
is  taken  in  the  so-called  "grammar  grades."  The 
mensuration  of  the  common  surfaces  and  solids  should, 
of  course,  never  be  a  matter  of  arbitrary  rule.  Our 
best  text-books  in  elementary  arithmetic  at  present 
give  satisfactory  development  of  the  rules  for  all 
necessary  cases  not  involving  irrational  numbers.  A 
pair  of  shears  and  some  cardboard  enable  the  teacher 
to  pass  from  the  rectangle  to  the  parallelogram,  and 
thence  to  the  trapezoid  and  the  triangle,  developing 
the  formulae  or  rules  with  little  difficulty.  Similarly 
the  formulae  for  the  circle  can  be  developed  by  cut- 
ting this  figure  into  sectors  which  are  approximately 

1  Rebifere,  Mathematiques  et  mathematicians,  p.  102. 


244     THE  TEACHING  OF  ELEMENTARY   MATHEMATICS 

triangles.  Only  a  little  labor  is  needed  to  prepare 
pasteboard  models  of  the  most  common  geometric 
solids,  and  these,  together  with  a  pail  of  dry  sand 
for  filling  some  of  them  in  comparing  volumes,  fur- 
S  nish  the  materials  for  developing  the  formulae  for 
measuring  such  bodies.1 

Nor  should  we  regard  this  method  of  investigation 
unscientific.  It  merely  follows  the  line  of  historic 
development,  the  line  in  which  truth  is  first  acquired 
by  induction.  Comte  cites  an  interesting  illustration 
of  this  method,  showing  the  way  in  which  Galileo 
determined  the  ratio  of  the  area  of  an  ordinary 
cycloid  to  that  of  the  generating  circle.  "  The  geome- 
try of  his  time  was  as  yet  insufficient  for  the  rational 
solution  of  such  problems.  Galileo  conceived  the 
idea  of  discovering  that  ratio  by  a  direct  experiment. 
Having  weighed  as  exactly  as  possible  two  plates  of 
the  same  material  and  of  equal  thickness,  one  of 
them  having  the  form  of  a  circle  and  the  other  that 
of  the  generated  cycloid,  he  found  the  weight  of  the 
latter  always  triple  that  of  the  former;  whence  he 
inferred  that  the  area  of  the  cycloid  is  triple  that  of 
the  generating  circle,  a  result  agreeing  with  the 
veritable  solution  subsequently  obtained  by  Pascal  and 

1  For  directions  as  to  this  work  see  Beman  and  Smith's  Higher  Arith- 
metic, Boston,  1896,  p.  66.  Reference  should  also  be  made  to  a  valuable 
pamphlet  by  Professor  Hanus,  Geometry  in  the  Grammar  School,  Boston, 
1893. 


WHAT  IS  GEOMETRY  24$ 

Wallis."  l  It  would  be  well,  indeed,  if  we  had  even 
more  of  this  induction  along  with  our  later  demonstra- 
tive geometry.  One  of  the  common  sources  of  failure, 
especially  in  the  discovery  of  loci  and  the  solution  of 
certain  other  problems,  is  the  inability  of  the  pupil  to 
make  correct  inductions  from  carefully  drawn  figures. 

Along  with  this  work  in  mensuration  should  con- 
tinue the  geometric  drawing  begun  in  the  earlier 
grades.  The  subject  has  been  worked  out  with  con- 
siderable success  by  several  writers.2 

Spencer's  Inventional  Geometry,  while  not  an  ideal 
text-book,  was  a  noteworthy  step  in  this  direction  of 
scientific  induction  based  upon  accurately  drawn  figures. 
Dr.  Shaw,  speaking  of  his  experiments  with  children 
along  the  lines  suggested  by  Spencer,  says :  "  A  few 
months'  work  proved  the  incalculable  value  of  inven- 
tional  geometry  in  a  school  course  of  study ;  and  eleven 
years'  experience  in  many  classes  and  in  different 
schools  confirms  that  judgment. 

"  In  these  classes  the  pleasure  experienced  in  the 
study  has  made  the  work  delightful  both  to  teacher  and 
to  taught;  and  there  has  always  been  a  continuous 

1  Philosophy  of  Mathematics,  English  by  Gillespie,  New  York,  1851, 
p.  1 86. 

2  Spencer,  W.  G.,  Inventional  Geometry,  New  York,  1876  ;    Harms, 
Erste  Stufe  des  mathematischen  Unterrichts,  II.  Abt.  3.  Aufl.,  Oldenburg, 
1878,  along  the  same  lines  as  a  work  by  Gille  (1854);  Schuster,  M.,  Auf- 
gaben  fur  den  Anfangsunterricht  in  der  Geometric,  Program,  Oldenburg, 
1897.     Campbell,  Observational  Geometry,  New  York,  1899. 


246    THE  TEACHING  OF  ELEMENTARY   MATHEMATICS 

interest  from  the  beginning  to  the  end  of  the  term. 
This  pleasure  and  interest  came,  not  from  any  environ- 
ment, not  from  the  peculiar  individuality  of  the  class, 
but  because  the  problems  are  so  graded  and  stated  that 
the  pupil's  progress  becomes  one  of  self -development  — 
a  realization  of  the  highest  law  in  education.  .  .  . 

"  The  pupil  should  not  be  told  or  shown,  but  thrown 
back  upon  himself;  for,  in  inventional  geometry,  the 
knowledge  is  to  be  gained  by  growth  and  experience, 
through  the  powers  he  possesses  and  the  method  of 
acquirement  peculiar  to  his  mind.  Occasionally  the 
pupil  is  not  a  little  baffled,  and  the  skill  of  the  teacher 
is  put  to  its  best  test  to  gain  the  solution  without  show- 
\  ing  or  telling  him.  Telling  or  showing  is  the  method  of 
the  instructor  —  not  the  teacher.  .  .  . 

"  Inventional  geometry  should  precede  the  demonstra- 
tive, so  as  to  give  the  pupil  many  concepts  to  draw 
upon  when  he  takes  up  syllogistic  demonstration.  De- 
monstrative geometry  then  becomes  an  easier  subject, 
and  he  is  surer  of  what  he  is  doing,  because  he  has 
more  general  notions  as  a  basis." 

Speaking  of  Spencer's  work,  Mr.  Langley,  one  of  the 
best  teachers  of  elementary  mathematics  in  England, 
confirms  the  views  already  expressed :  "  It  has  not  been 
usual  for  students,  at  any  rate  in  schools,  to  approach 
the  study  of  geometry  in  this  experimental  way,  though 
there  have  probably  always  been  individual  teachers 
who  have  used  it  to  varying  extents.  Of  late  years, 


WHAT  IS  GEOMETRY  247 

however,  —  in  fact  since  more  attention  has  been  given 
to  the  theory  and  practice  of  education,  —  it  has  been 
strongly  advocated.  My  own  experience  confirms  me 
day  by  day  in  the  opinion  that  it  is  the  best  method 
for  the  majority  of  students,  though  a  few  may  be 
able  to  dispense  with  it. 

"It  has  two  advantages:  (i)  It  leads  to  clear  con- 
ceptions  of  the  truths  to  be  established ;  (2)  it  may  be 
used  to  introduce  the  student  naturally  to  a  different 
method  of  establishing  such  truths  —  the  deductive 
method."  * 

In  America  Professor  Hanus  has  been  prominent 
in  putting  the  work  on  a  practical  basis.2  He  rec- 
ommends two  recitation  periods  per  week  for  the 
seventh  and  eighth  grades,  and  one  for  the  ninth, 
the  periods  to  be  at  least  thirty  minutes  long.  The 
following  are  his  guiding  principles  for  teachers : 

"  I.  Early  instruction  in  geometry  should  be  ob- 
ject teaching. 

"2.  The  pupil  should  make  and  keep  an  accurate 
record  of  his  observations,  and  of  the  definitions  or 

1  Langley,  E.  M.,  How  to  learn  Geometry,  The  Educational  Review 
(London),  Vol.  VIII,  O.  S.,  p.  3.    The  subject  is  also  discussed,  with  a  brief 
list  of  German  text-books,  in  Dressler's  Der  mathematisch-naturwissen- 
schaftliche  Unterricht  an   deutschen  (Volksschullehrer-)   Seminaren,  in 
Hoffmann's  Zeitschrift,  XXIII.  Jahrg.,  p.  18. 

2  Outline  of  work  in  Geometry  for  the  Seventh,  Eighth,  and  Ninth 
Grades  of  the  Cambridge  Public  Schools,  Boston,  1893;  Geometry  in  the 
Grammar  School,  Boston,  1893. 


248    THE  TEACHING  OF  ELEMENTARY   MATHEMATICS 

propositions  which  his  examination  of  the  object  or 
objects  has  developed. 

"3.  In  all  his  work  the  pupil  should  be  taught 
to  express  himself  by  drawing,  by  construction,  and 
in  words,  as  fully  and  accurately  as  possible.  The 
language  finally  accepted  by  the  teacher  should  be 
the  language  of  the  science,  and  not  a  temporary 
phraseology  to  be  set  aside  later. 

"4.  The  pupil  is  to  convince  himself  of  geomet- 
rical truths  primarily  through  measurement,  drawing, 
construction,  and  superposition,  not  by  a  logical  dem- 
onstration. But  gradually  (especially  during  the  last 
year  of  the  work)  the  pupil  should  be  led  to  attempt 
the  general  demonstration  of  all  the  simpler  propo- 
sitions. 

"  5.  The  subject  should  be  developed  by  the  com- 
bined effort  of  teacher  and  pupil,  i.e.  the  teacher 
and  the  pupil  are  to  cooperate  in  reconstructing  the 
subject  for  themselves.  This  is  best  accomplished  by 
skilful  questioning  without  the  use  of  a  text-book  con- 
taining the  definitions,  solutions,  and  demonstrations.  .  .  . 

"6.  The  subject-matter  of  each  lesson  should  be 
considered  in  its  relation  to  life,  i.e.  the  actual 
occurrence  in  nature  and  in  the  structures  of  ma- 
chines made  by  man  of  the  geometrical  forms  studied, 
and  the  application  of  the  propositions  to  the  ordi- 
nary affairs  of  life  should  be  the  basis  and  the 
outcome  of  every  exercise.  .  .  . 


WHAT  IS  GEOMETRY  249 

"7.  Accuracy  and  neatness  are  absolutely  essen- 
tial in  all  work  done  by  the  pupils."1 

In  Germany  a  course  extending  through  what 
corresponds  to  our  "  grammar  school "  has  been  out- 
lined by  several  writers.  Without  going  into  details, 
the  following  course  suggested  by  Rein  may  serve 
to  show  what  ground  the  modern  Herbartians  pro- 
pose to  cover. 

A.  Geometric  form  (Geometrische  Formenlehre). 

Fourth  school  year — The  cube,  square  prism,  ob- 
long prism,  triangular  prism,  quadrangular  pyramid. 
In  addition  to  these  solids  the  pupil  considers  the 
point,  straight  line,  surface,  direction,  measurement  of 
the  straight  line,  the  right  angle  and  its  parts,  the 
square  and  its  construction,  the  rectangle  and  its 
construction,  the  triangle,  and  the  diagonals  .of  the 
rectangle. 

Fifth  school  year — The  hexagonal  prism,  octagonal 
prism,  hexagonal  and  octagonal  pyramid,  truncated 
pyramid,  cylinder,  cone,  truncated  cone,  and  sphere. 
The  following  plane  figures  are  also  studied :  the 
regular  hexagon  and  octagon,  the  obtuse  angle,  the 
trapezoid  and  circle. 

B.  Geontetry. 

Sixth  school  year —  Properties  of  magnitudes  (Eigen- 
schaften,  Gesetze,  der  Raumgrossen),  constructions,  and 
mensuration.  Size  and  measurement  of  angles,  the 

1  Hanus.    The  course  is  outlined  in  both  pamphlets. 


250    THE  TEACHING  OF  ELEMENTARY  MATHEMATICS 

protractor.  Division  of  angles.  Kinds  and  properties 
of  triangles  and  parallelograms,  with  constructions. 
Mensuration  of  surfaces,  the  square,  rectangle,  paral- 
lelogram, and  triangle.  The  trapezoid.  The  circle, 
its  sectors  and  segments,  and  the  value  of  TT.  Reg- 
ular polygons. 

Seventh  school  year — Measurement  and  drawing  of 
solids. 

C.  Practical  geometry. 

Eighth  school  year —  i.  The  congruence  proposi- 
tions. 2.  Similarity.  3.  Pythagorean  theorem.  Appli- 
cations to  practical  mensuration. J 

Demonstrative  geometry  —  The  next  step  brings  the 
student  to  demonstrative  geometry,  the  geometry  of 
Euclid,  or  its  equivalent.  Here  the  educator  is  at 
once  confronted  by  the  question,  When  shall  this 
work  be  begun  ? 

In  England  Euclid  is  begun  at  an  age  which  sur- 
prises American  educators.  In  the  lyce"es  of  France 
and  the  Gymnasien  (or  Realschulen,  etc.)  of  Germany, 
as  well  as  in  most  of  the  other  preparatory  schools 
of  Europe,  demonstrative  geometry,  although  not 
Euclid,  also  finds  much  earlier  place  than  in  America. 
With  us  the  subject  usually  begins  in  the  tenth  or 
eleventh  school  year,  and  the  "Committee  of  Ten" 
recommends  no  change  in  this  plan.  To  begin  a 

1  Rein,  Pickel  and  Scheller,  Theorie  und  Praxis  des  Volksschulunter- 
richts;  Das  vierte  Schuljahr,  3.  Aufl.,  Leipzig,  1892,  p.  232. 


WHAT  IS  GEOMETRY  25 1 

work  of  the  difficulty  of  Euclid  any  earlier  than  this 
will  hardly  be  sanctioned  by  American  teachers;  the 
hard  Euclidean  method  must  change,  or  the  subject 
must  remain  thus  late  in  the  curriculum.  If  the 
object  were,  as  seems  to  be  the  case  in  England, 
to  cram  the  memory  for  an  examination,  it  could  be 
attained  here  as  easily  as  there.  But  the  considerable 
personal  experience  of  the  writer,  as  well  as  the  far 
more  extended  researches  of  others,  convinces  him 
that  as  a  valuable  training  in  logic,  as  a  stimulus  to 
mathematical  study,  and  as  a  foundation  for  future 
research,  the  study  of  Euclid  as  undertaken  in  Eng- 
land is  not  a  success. l  If  one  has  any  doubt  as  to 
this  judgment,  it  should  be  removed  by  this  recent 
testimony  of  Professor  Minchin,  a  man  thoroughly 
familiar  with  the  system,  and  an  excellent  math- 
ematician and  teacher  in  spite  of  the  fact  that  he 
was  brought  up  on  Euclid. 

"  Why,  then,"  he  says,  "  is  it  that  the  teacher,  when 
he  comes  to  the  teaching  of  Euclid,  is  confronted 
with  such  great  difficulties  that  his  belief  in  the 
rationality  of  human  beings  almost  disappears  with 
the  last  vestiges  of  that  good  temper  which  he  him- 
self once  possessed?  The  reason  is  simply  that 

1  Holzmflller,  G.,  Notwendigkeit  eines  propadeutisch-mathematischen 
Unterrichts  in  den  Unterklassen  hoherer  Lehranstaltcn  vor  dem  wis- 
senschaftlich-systematischen,  Hoffmann's  Zeitschrift,  XXVI.  Jahrg.,  p.  321, 
334- 


252    THE  TEACHING  OF  ELEMENTARY   MATHEMATICS 

Euclid's  book  is  not  suitable  to  the  understanding  of 
young  boys.  It  fails  signally  as  regards  both  its  lan- 
guage and  its  arrangement.  .  .  .  For  myself,  I  con- 
fess that,  to  the  best  of  my  belief,  I  had  been  through 
the  six  books  of  Euclid  without  really  understanding 
the  meaning  of  an  angle  "^ 

If,  however,  a  series  of  text-books  should  appear 
which  carried  the  essential  part  of  the  first  three 
books  of  Euclid  along  with  the  arithmetic  and  alge- 
bra work  of  the  seventh,  eighth,  and  ninth  school 
years,  thus  connecting  the  severe  demonstrative  ge- 
ometry with  that  outlined  for  the  lower  grades,  it 
would  then  be  entirely  feasible  to  begin  demonstra- 
tive geometry  earlier  than  now.  We  have,  however, 
no  such  books  in  English,  at  least  none  which  have 
succeeded  in  any  such  way  as  Holzmiiller's  excel- 
lent series  has  in  Germany.2  That  a  child  in  the 
seventh  grade  can  handle  the  pons  asinorum  of 
Euclid  quite  as  easily  as  the  problems  he  often 
meets  in  arithmetic,  has  been  shown  too  often  to 
admit  of  dispute.  But  in  America  we  have  been 
showing  this  only  in  sporadic  cases,  without  formu- 
lating a  well-ordered  scheme  of  work  which  should 
spread  the  geometry  out,  along  with  the  algebra  and 
the  arithmetic.  It  is  reasonable  to  expect  that  this 

1  The  School  World  (London),  Vol.  I,  1899,  p.  161. 

2  In  this  connection  the  conclusion  of  Holzmuller's  article  mentioned 
on  p.  251  is  of  interest. 


WHAT  IS  GEOMETRY  253 

plan  will  materialize  before  many  years,  through  the 
skilful  labors  of  some  educated  writer  of  a  series 
of  text-books.  "That  algebra,  arithmetic,  and  geome- 
try should  be  taught  side  by  side  is  not  merely  use- 
ful; it  is  indispensable  for  maintaining  that  unity 
and  coordination  in  mathematics,  without  which  the 
science  loses  all  interest  and  value.  A  boy  who  has 
taken  his  arithmetic  first,  and  then  his  algebra,  and 
then  his  geometry,  has  his  mental  powers  less  de- 
veloped (I* esprit  moins  form/)  than  they  would  have 
been  with  three  or  four  years  of  parallel  teaching 
intelligently  pursued." * 

Naturally  a  child  loves  form  quite  as  much  as 
number.  Practically  he  needs  number  more  often, 
and  hence  the  elements  of  computation  have  been 
taught  to  him  at  an  early  age.  But  when  we  come 
into  the  theoretical  part  of  arithmetic  —  greatest  com- 
mon divisor,  roots,  proportion,  etc.  —  it  is  merely  an 
accident  (historically  explainable)  that  education  has 
carried  the  child  to  the  study  of  number  and  func- 
tions rather  than  to  the  study  of  form. 

Hence  in  general  it  may  be  said  that  the  study  of 
demonstrative  geometry  might  profitably  begin  earlier 
than  it  does  in  the  American  schools,  but  that  this 
would  require,  for  the  best  results,  a  style  of  presen- 
tation quite  different  from  that  of  Euclid  or  his 
modern  followers. 

1  Laisant,  La  Mathematique,  p.  227. 


254    THE  TEACHING  OF  ELEMENTARY  MATHEMATICS 

The  use  of  text-books  —  But  taking  the  curriculum 
as  it  stands  in  America  at  present,  what  general 
method  of  presentation  shall  be  followed,  and  what 
kind  of  text-book  shall  be  recommended?  The  great 
majority  of  teachers  take  some  text-book,  require  the 
pupils  to  prove  the  theorems  substantially  as  therein 
set  forth,  and  demand  the  demonstration  of  a  con- 
siderable number  of  propositions  which  the  English 
call  "riders"  —  a  name  quite  as  good  (and  bad)  as 
our  "  original  exercises."  The  result  is  a  tendency  to 
fall  into  the  habit  of  merely  memorizing  the  solutions, 
thus  losing  sight  of  the  greatest  value  of  the  subject 
—  the  training  which  it  gives  in  logic. 

To  avoid  this  danger,  numerous  plans  have  been 
devised.  One  is  that  of  dictating  the  propositions, 
giving  a  few  suggestions,  and  requiring  the  pupil  to 
work  out  his  own  proofs.  This  plan,  however,  is 
open  to  several  objections  so  serious  as  to  condemn 
it  in  the  minds  of  most  educators.  In  the  first  place 
there  is  a  great  waste  of  time  in  the  dictation  of 
notes  —  a  return  to  medievalism.  Furthermore,  if  the 
usual  sequence  of  propositions  is  varied,  few  teachers 
have  the  ability  to  make  this  variation  without  destroy- 
ing something  of  the  logic  or  symmetry  of  the  sub- 
ject ;  if  the  usual  sequence  is  followed,  the  pupil  simply 
secures  some  text-book  on  geometry,  often  a  poor  one, 
and  memorizes  from  that.  Again,  the  pupil  loses  the 
advantage  of  having  constantly  before  him  a  standard 


WHAT  IS  GEOMETRY  255 

of  excellence  in  logic,  in  drawing,  and  in  arrangement 
of  work,  and  he  fails  to  acquire  the  power  to  read 
and  assimilate  mathematical  literature,  a  serious  bar 
to  his  subsequent  progress  in  more  advanced  lines. 

To  meet  the  first  of  the  above  objections,  the  waste 
of  time  in  dictation,  text-books  have  been  prepared 
containing  merely  the  definitions,  postulates,  axioms, 
enunciations,  etc.  But  while  free  from  the  first  objec- 
tion, they  are  open  to  the  others,  and  hence  have  met 
with  only  slight  favor. 

Text-books  have  also  been  prepared  which,  in  place 
of  the  proofs,  submit  series  of  questions,  the  answers 
to  which  lead  to  the  demonstrations.  This  is  the 
heuristic  method  put  into  book  form;  it  substitutes  a 
dead  printed  page  for  a  live  intelligent  teacher.  The 
substitution  is  necessarily  a  poor  one,  for  the  printed 
questions  usually  admit  of  but  a  single  answer  each, 
and  hence  they  merely  disguise  the  usual  formal  proof. 
They  give  the  proof,  but  they  give  no  model  of  a  logical 
statement. 

The  kind  of  text-book  which  the  world  has  found 
most  usable,  and  probably  rightly  so,  is  that  which 
possesses  these  elements:  (i)  A  sequence  of  proposi- 
tions which  is  not  only  logical,  but  psychological ;  not 
merely  one  which  will  work  theoretically,  but  one  in 
which  the  arrangement  is  adapted  to  the  mind  of  the 
pupil;  (2)  Exactness  of  statement,  avoiding  such  slip- 
shod expressions  as,  "A  circle  is  a  polygon  of  an  in- 


256    THE  TEACHING  OF  ELEMENTARY   MATHEMATICS 

finite  number  of  sides,"  "  Similar  figures  are  those  with 
proportional  sides  and  equal  angles,"  without  other 
explanation;  (3)  Proofs  given  in  a  form  which  shall 
be  a  model  of  excellence  for  the  pupil  to  pattern  after ; 

(4)  Abundant  exercises  from  the  beginning,  with  prac- 
tical  suggestions   as   to    methods   of    attacking    them; 

(5)  Propaedeutic  work  in  the  form  of  questions  or  exer- 
cises,  inserted    long   enough    before    the    propositions 
concerned  to  demand  thought  —  that  is,   not  immedi- 
ately preceding  the  author's  proof. 

Such  a  book  gives  the  best  opportunity  for  success- 
ful work  at  the  hands  of  a  good  instructor.  But  no 
book  can  ever  take  the  place  of  an  enthusiastic,  re- 
sourceful teacher.  In  the  hands  of  a  dull,  mechanical, 
gradgrind  person  with  a  teacher's  license,  no  book 
can  be  successful.  The  teacher  who  does  not  antici- 
pate difficulties  which  would  otherwise  be  discouraging 
to  the  pupil,  tempering  these  difficulties  (but  not  wholly 
removing  them)  by  skilful  questions,  is  not  doing  the 
best  work.  On  the  other  hand,  the  teacher  who  over- 
develops, who  seeks  to  eliminate  all  difficulties,  who 
does  all  of  the  thinking  for  the  class,  is  equally  at 
fault.  Youth  takes  little  interest  in  that  which  offers 
no  opportunity  for  struggle,  whether  it  be  on  the  play- 
ground, in  the  home  games  of  an  evening,  or  in  the 
classroom. 


CHAPTER   XI 
THE  BASES  OF  GEOMETRY 

The  bases  —  Geometry  as  a  science  starts  from  cer- 
tain definitions,  axioms,  and  postulates.  It  is  hardly 
the  province  of  this  work  to  enter  into  a  philosophi- 
cal discussion  of  the  foundations  upon  which  the 
science  rests,  first  because  such  a  discussion  would 
require  a  volume  of  some  size,1  and  also  because 
practically  the  teacher  is  unable  materially  to  change 
the  definitions,  axioms,  and  postulates  of  the  text- 
book which  happens  to  be  in  the  hands  of  his 
pupils.  A  brief  consideration  of  these  bases  of  the 
science  may,  however,  be  of  service. 

The  definitions  of  geometry  occupy  a  position  some- 
what different  from  that  held  by  the  definitions  of 
algebra  and  arithmetic.  There  is  not  the  same 
necessity  for  exactness  in  the  definition  of  monomial 

1  The  teacher  may  consult  Dixon,  E.  T.,  The  Foundations  of  Geometry, 
Cambridge,  1891  ;  Russell,  An  Essay  on  the  Foundations  of  Geometry, 
Cambridge,  1897;  Poincare,  On  the  Foundations  of  Geometry,  The 
Monist,  October,  1898  ;  Hilbert,  D.,  Grundlagen  der  Geometric,  in  Fest- 
schrift zur  Feier  der  Enthiillung  des  Gauss-Weber-Denkmals  in  Gottingen, 
Leipzig,  1899;  Veronese,  G.,  Fondamenti  di  Ge^metria,  Padova,  1891; 
Koenigsberger,  L.,  Fundamental  Principles  of  Mathematics,  Smithsonian 
Report,  1896,  p.  93. 

s  257 


258    THE  TEACHING  OF  ELEMENTARY  MATHEMATICS 

as  in  that  of  right  angle,  for  the  latter  is  a  control- 
ling factor  in  several  logical  demonstrations,  while  the 
former  is  not.  In  the  same  way  more  care  must  be 
shown  in  the  definition  of  similar  figures  than  in  that 
of  simultaneous  equations,  of  isosceles  triangle  than  of 
incomplete  quadratic,  of  parallelepiped  than  of  binomial ; 
not  that  all  of  these  terms  must  not  be  well  under- 
stood and  properly  used,  and  not  that  algebra  is  less 
exact  than  geometry,  but  that  the  geometric  terms 
enter  into  logical  proofs  in  such  way  as  to  make  their 
exact  statement  a  matter  of  greater  moment. 

Hence  the  suggestions,  already  made  in  Chapter  VIII 
upon  accuracy  of  definition  in  algebra,  apply  with  even 
greater  force  to  geometry.  Nor  should  the  teacher 
attend  so  much  to  the  idea  that  all  the  truth  cannot 
be  taught  at  once,  as  to  acquire  the  dangerous  habit 
of  teaching  partial  truths  only,  or  (as  too  often  happens) 
of  teaching  mere  words,  sometimes  unintelligible,  some- 
times wholly  false.  A  few  selections  from  our  elemen- 
tary text-books  will  illustrate  these  points. 

We  often  see,  for  example,  as  a  definition,  "A 
straight  line  is  the  shortest  distance  between  two 
points."  Now  in  the  first  place  this  is  absurd,  be- 
cause a  line  is  not  distance ;  distance  is  measured 
on  a  line,  and  usually  on  a  curved  one.  Further- 
more, the  statement  merely  gives  one  property  of  a 
straight  line;  it  is  a  theorem,  and  by  no  means  an 
easy  one  to  prove.  A  definition  should  be  stated  in 


THE  BASES  OF  GEOMETRY  259 

terms  more  simple  than  the  term  defined ;  but  distance 
is  one  of  the  most  difficult  of  the  elementary  con- 
cepts to  define.1  Mathematicians  have  long  since 
abandoned  the  statement.  "  It  is  a  definition  almost 
universally  discarded,  and  it  represents  one  of  the 
most  remarkable  examples  of  the  persistence  with 
which  an  absurdity  can  perpetuate  itself  through  the 
centuries.  In  the  first  place,  the  idea  expressed  is 
incomprehensible  to  beginners,  since  it  presupposes 
the  idea  of  the  length  of  a  curve;  and  further,  it  is  a 
case  of  reasoning  in  a  circle  (c'est  un  cercle  vicieux), 
for  the  length  of  a  curve  can  only  be  understood 
as  the  limit  of  a  sum  of  rectilinear  lengths.  And 
finally,  it  is  not  a  definition  at  all,  but  rather  a 
demonstrable  proposition."2 

The  fact  is,  the  concept  straight  line  is  element- 
ary; it  is  not  capable  of  satisfactory  definition,  and 
hence  it  should  be  given  merely  some  brief  explana- 
tion. From  Plato's  time  to  our  own,  attempts  have 
been  made  to  define  such  fundamental  concepts  as 
straight  line  and  angle,  but  with  no  success.  As 

1  Pascal's  rules  for  definitions  are  worthy  of  consideration  in  this 
connection  :  "(i)  Do  not  attempt  to  define  any  terms  so  well  known  in 
themselves  that  you  have  no  clearer  terms  by  which  to  explain  them  ; 

(2)  Admit  no  terms  which  are  obscure  or  doubtful,  without  definition ; 

(3)  Employ  in  definitions   only  terms  which   are   perfectly  well  known 
or  which   have   already  been   explained."      Rebiere,   Mathlmatiques  et 
mathematiciens,  p.  23. 


OF  THE 

.  UNIVERSITY 

\  OF 


260     THE  TEACHING  OF  ELEMENTARY  MATHEMATICS 

St.  Augustine  said  of  time,  "  If  you  ask  me  what 
it  is,  I  cannot  tell  you;  but  if  you  do  not  ask  me, 
I  know  too  well."  And  Pascal  said  of  geometry :  "It 
may  be  thought  strange  that  geometry  is  unable  to 
define  any  of  its  principal  concepts;  for  it  cannot 
define  movement,  or  number,  or  space,  and  yet  these 
are  the  very  things  which  it  considers  most.  It  is 
not  surprising,  however,  when  we  consider  that  this 
admirable  science  attaches  itself  only  to  the  most 
simple  concepts,  and  that  the  very  quality  which 
makes  these  worthy  of  being  its  objects  renders  them 
incapable  of  definition.  Hence  the  inability  to  de- 
fine is  rather  a  merit  than  a  defect,  since  it  arises 
not  from  the  obscurity  of  the  concepts,  but  rather 
from  their  extreme  evidence."1 

Text-books  are  also  liable  to  err  on  the  side  of 
redundancy  in  definition,  as  in  the  statement,  "  A 
rectangle  is  a  parallelogram  all  of  whose  angles  are 
right  angles."  It  would  be  thought  absurd  to  say, 
"A  rectangle  is  a  four-sided  parallelogram  whose  op- 

1  Rebiere,  Mathematiques  et  mathematicians,  p.  16.  For  those  who 
wish  thoroughly  to  investigate  the  matter  of  the  elementary  definitions 
(straight  line,  angle,  etc.),  it  will  be  of  value  to  know  that  Schotten  has 
compiled  all  of  the  typical  definitions  of  these  concepts  which  have  ap- 
peared from  the  time  of  the  Greeks  to  the  present,  and  has  set  them  forth 
with  critical  notes  in  his  valuable  treatise,  Inhalt  und  Methode  des  plani- 
metrischen  Unterrichts,  Bd.  I,  1890;  Bd.  II,  1893;  Bd-  HI,  in  press. 
Professor  Newcomb  has  also  considered  the  matter  briefly  in  the  Appendix 
to  his  Geometry. 


THE  BASES  OF  GEOMETRY  261 

posite  sides  are  equal  and  parallel,  and  all  of  whose 
angles  are  right  angles,"  because  of  the  manifest 
redundancy.  But  if  the  definition  is  given  at  the 
proper  place,  it  suffices  to  say,  "  If  one  angle  of  a 
parallelogram  is  a  right  angle,  the  parallelogram  is 
called  a  rectangle."  The  same  criticism  applies  to 
one  of  the  common  definitions  of  a  square,  "A  rec- 
tangle whose  sides  are  all  equal " ;  it  suffices  if  two 
adjacent  sides  are  equal.  The  definition  commonly 
given  of  similar  figures  is  an  illustration  of  the  teach- 
ing of  a  half  truth,  the  whole  truth  being  thereby 
permanently  excluded,  and  all  this  with  no  excuse. 
If  a  student  beginning  geometry  were  asked  to  name 
two  similar  figures,  he  would  probably  suggest  two 
circles,  or  two  spheres,  or  two  straight  lines,  or  two 
squares,  and  he  would  be  right.  But  when  he  comes 
to  the  definition  he  finds  that,  of  the  four  classes  of 
figures  named,  only  the  squares  are  similar.  It  is, 
however,  an  easy  matter  to  define  similar  systems  of 
points,  and  then  to  say,  "Two  figures  are  said  to 
be  similar  when  their  systems  of  points  are  similar," 
thus  admitting  circles,  spheres,  similar  cones,  etc.,  all  / 
of  which  are  excluded  by  the  usual  text-book  defini- 
tion, and  all  of  which  deserve  to  be  considered.1 

The  introduction  of  the  modern  chapter  on  maxima 
and  minima,  in  many  of  our  elementary  works,  makes 

1  For  further  discussion  see  Beman  and  Smith's  New  Plane  and  Solid 
Geometry,  Boston,  1899,  p.  182. 


262     THE  TEACHING  OF  ELEMENTARY  MATHEMATICS 

it  worth  while  to  say  that  the  definition  of  maximum 
as  the  greatest  value  a  variable  can  take,  not  only 
is  misleading  at  the  time,  but  also  is  conducive  to  sub- 
sequent misunderstanding.  Every  teacher  of  geom- 
etry must  be  aware  that,  in  general,  a  variable  may 
have  several  maxima. 

The  laxness  of  definitions  which  creeps  into  ele- 
mentary work  is  well  illustrated  in  the  case  of  the 
polyhedral  angle.  We  not  unfrequently  find  angle 
defined  as  "the  difference  of  direction  between  two 
lines  which  meet"  (a  poor  definition  because  the  word 
angle  is  quite  as  elementary  as  the  word  direction), 
and  the  polyhedral  angle  defined  as  "the  angle 
formed  by  three  or  more  planes  meeting  in  a  point." 
The  absurdity  appears  when  we  substitute  the  defi- 
nition of  angle  for  the  word:  "A  polyhedral  angle 
is  '  the  difference  of  direction  between  two  lines  which 
meet'  formed  by  three  or  more  planes,"  etc.,  and  yet 
we  teach  mathematics  as  an  exact  science !  This  illus- 
tration is  not  a  "man  of  straw";  one  need  not  look 
far  to  find  it. 

Axioms  and  postulates  —  In  considering  briefly  the 
nature  and  the  r61e  of  the  axioms  and  postulates  of 
geometry,  we  may  well  begin  by  asking  the  meaning 
of  the  terms  themselves. 

Of  course  it  is  true  that  these  words  mean  to  any 
generation  just  what  the  world  at  that  time  agrees 
they  shall  mean,  and  hence  it  is  not  a  valid  argu- 


THE  BASES  OF  GEOMETRY  263 

ment  to  say  that  Euclid  did  not  employ  them  in 
the  sense  understood  by  his  early  English  trans- 
lators. At  the  same  time  there  has  been,  for  a 
number  of  years,  a  feeling  that  the  common  defini- 
tions of  postulate  and  axiom  are  absurd  in  statement 
and  unscientific  in  thought,  as  well  as  unjustifiable 
historically.  Heiberg,1  the  most  scholarly  editor  of 
the  Elements,  has  considered  the  matter  very  thor- 
oughly, and  is  convinced  that  Euclid  used  axiom  for 
a  general  mathematical  truth  accepted  without  proof, 
and  postulate  for  a  similar  truth  of  a  geometric 
nature.  Thus  the  statement,  "  If  equals  are  added 
to  equals  the  sums  are  equal,"  is  an  axiom ;  but, 
"Through  a  given  point  but  one  line  can  be  drawn 
parallel  to  a  given  line,"  is  a  postulate  (not,  how- 
ever, in  Euclid's  language).  The  notion  that  an 
axiom  is  a  "self-evident  theorem,"  and  a  postulate 
a  problem  too  simple  for  solution,  is  therefore  his- 
torically incorrect,  as  well  as  absurd  in  substance.  A 
return  to  Euclid's  use  of  the  words  would  seem  desir- 
able, although  the  single  word  axiom  for  both  classes 
would  simplify  matters. 

The  definition  of  axiom  as  "  a  self-evident  truth  "  has 
already  been  characterized  as  absurd.  For  what  is  self- 
evident  to  one  mind  is  not  at  all  so  to  another.  It  may 
be  "  self-evident "  to  a  very  good  student  that  i  is  the 
only  number  whose  cube  is  i,  until  he  tries  cubing 

1  Euclidis  elementa,  Leipzig,  1883-88. 


264    THE  TEACHING  OF  ELEMENTARY   MATHEMATICS 

—  J  ±  jV  — 3;  or  that  2  is  the  only  fourth  root  of  16, 
until  some  one  suggests  three  others ;  or  that  ab  must 
always  equal  ba,  until  he  studies  quaternions  or  the 
theory  of  groups.  The  fact  is,  in  geometry  the  word 
"axiom"  is  used  merely  to  designate  certain  general 
statements  the  truth  of  which  is  assumed.  Our  senses 
seem  to  indicate  that  they  are  true ;  but  whether  true  or 
false,  we  take  them  for  granted  and  see  whither  they 
lead  us. 

Similarly,  in  geometry,  with  the  word  "postulate."  A 
postulate  is  a  statement,  referring  to  geometry,  the  truth 
of  which  is  assumed.  The  statement  may  be  true  or  it 
may  be  false,  although  our  senses  seem  to  indicate  the 
former.  That  space  is  homogeneous  seems  true,  but  it 
may  not  be ;  but  we  assume  it  true  and  see  whither  we 
are  led.  So  we  may  be  able  to  draw,  through  a  given 
point,  more  than  one  line  parallel  to  a  given  line,  although 
our  senses,  especially  as  biassed  by  our  early  training, 
seem  to  indicate  not.  But  any  one  is  entirely  at  liberty 
to  deny  this  or  any  other  postulate,  and  to  build  up  a 
logical  geometry  accordingly,  if  he  can.  In  the  case  of 
the  postulate  of  parallel  lines  this  was  done  by  Loba- 
chevsky  and  Bolyai,  and  their  geometries  are  entirely 
logical.1  Mathematicians  generally  agree  that  the  post- 

l¥or  references,  Smith,  D.  E.,  History  of  Modern  Mathematics, 
p.  565.  The  best  historical  treatment  of  the  subject  is  that  by  Stackel 
and  Engel,  Die  Theorie  der  Parallellinien  von  Euklid  bis  auf  Gauss 
Leipzig,  1895. 


THE  BASES  OF  GEOMETRY  265 

ulate  is  not  at  all  "  self-evident."  As  Klein,  the  well- 
known  Gottingen  professor,  says,  "As  mathematicians 
we  must  array  ourselves  as  opponents  of  Kant's  idea 
that  the  parallel  axiom  is  to  be  considered  an  a  priori 
truth." 1  Lobachevsky  and  Bolyai  postulate  that  through 
a  given  point  more  than  one  line  can  be  drawn  parallel 
to  a  given  line,  and  on  this,  together  with  most  of  the 
axioms,  postulates,  and  definitions  of  Euclid,  they  build 
up  a  perfectly  consistent  geometry. 

Similarly,  as  in  plane  geometry  we  postulate  that 
space  has  three  dimensions  and  that  a  plane  figure  may 
be  revolved  about  an  axis,  through  three-dimensional 
space,  so  as  to  coincide  with  a  symmetric  figure,  so  in 
solid  geometry  we  might  postulate  that  a  solid  may  be 
revolved  through  a  four-dimensional  space  so  as  to 
coincide  with  a  symmetric  solid,  e.g.,  a  right-hand  glove 
with  a  left-hand  one.  A  perfectly  consistent  geometry 
could  be  constructed  with  this  as  a  postulate.2 

A  postulate  is  not,  therefore,  a  "self-evident"  state- 
ment; it  is  a  geometric  assumption.  In  ordinary  ele- 
mentary geometry  we  postulate  only  certain  relations 
which  most  people  are  willing  to  say  agree  with  their 
sense-perceptions.  They  do  not  entirely  agree  with 
them,  for  we  have  no  sense-perception  of  a  straight 

1  Vergleichende  Betrachtungen,  Erlangen,  1872. 

2  For  a  brief  and  popular  statement  concerning  the  fourth  dimension, 
see  the  recent  translation  of  Schubert,  H.f  Mathematical  Essays  and  Rec- 
reations, Chicago,  1898,  p.  64. 


266    THE  TEACHING  OF  ELEMENTARY  MATHEMATICS 

line,  nor,  a  fortiori,  of  two  parallels.  Our  geometric 
concepts  are  all  abstractions  made  from  our  physical 
concepts.1  As  D' Alembert  says,  "  Geometric  truths 
are  a  kind  of  asymptote  of  physical  truths,  i.e.,  the 
limit  which  they  indefinitely  approach  without  ever 
exactly  reaching." 

As  to  the  number  of  postulates  or  axioms,  the  ques- 
tion is  wholly  unsettled.  Practically,  the  teacher  of 
the  elements  will  follow  those  given  in  his  text-book. 
But  as  has  been  truly  said,  the  list  usually  given  is 
both  insufficient  and  superabundant,  since  on  the  one 
hand  we  use  postulates  not  laid  down  in  the  ordinary 
text-books,  and  on  the  other  hand  we  can  demon- 
strate some  of  those  which  are  given,  so  that  it  is 
unnecessary  to  assume  them.2 

The  most  recent  examination  of  the  postulates  of 
rectilinear  figures  is  that  of  Hilbert,3  and  is  here  set 
forth  in  some  detail  because  of  the  high  mathemati- 
cal authority  with  which  it  comes  to  us.  "  In  geome- 
try we  consider  three  different  systems  of  things. 


1  Les  figures  geometriques  sont  de  pures  conceptions  de  1'esprit.    Com- 
pagnon. 

2  De  Tilly,  in  Rebiere,  Les  Mathematiques,  etc.,  p.  21.     He  adds,  "  The 
axioms  of  geometry  can  be  reduced  to  three,  that   of  distance   and  its 
essential  properties,  that  of  the  indefinite  increase  of  distance,  and  that 
of  unique  parallelism." 

8  Hilbert,  D.,  Grundlagen  der  Geometric,  in  the  Gauss- Weber-Denk- 
mals  Festschrift,  Leipzig,  1899.  See  the  author's  review  in  The  Educa- 
tional Review,  January,  1900. 


THE  BASES  OF  GEOMETRY  267 

The  things  of  the  first  system  we  call  points,  desig- 
nating them  A,  B,  C,  •-  ;  the  things  of  the  second 
system  we  call  straight  lines,  designating  them  a,  b, 
ct  ••• ;  the  things  of  the  third  system  we  call  planes, 
designating  them  a,  $,  7,  •••.  The  points  we  may 
call  the  elements  of  linear  geometry;  the  points  and 
straight  lines  the  elements  of  plane  geometry;  the 
points,  straight  lines,  and  planes  the  elements  of 
spatial  geometry  or  of  space. 

"We  consider  the  points,  lines,  and  planes  in  cer- 
tain mutual  relations,  and  we  designate  these  relations 
by  the  words,  'lie,'  'between/  'parallel,'  'congruent,' 
'  continuous,'  and  the  exact  and  complete  description  of 
these  relations  follows  from  the  axioms  of  geometry. 

"These  axioms  separate  into  five  groups,  each  ex- 
pressing certain  fundamental  facts  of  our  conscious- 
ness :  — 

"I.    Axioms  of  connection  (Verkniipfung). 

"  i.  Two  different  points,  A,  B,  determine  a  straight 
line  a,  and  we  say  that  AB  =  a,  or  BA  =  a.1 

"  2.  Any  two  different  points  on  a  straight  line  de- 
termine that  line;  i.e.t  if  AB  —  a  and  AC =  a,  and 
B  is  not  C,  then  BC  =  a. 

"3.  Three  non-collinear  points,  A,  B,  C,  determine 
a  plane  «,  and  we  say  that  ABC  =  «. 

"4.  Any  three  non-collinear  points,  A,  B,  Ct  of  a 
plane  «,  determine  a. 

1  Of  course  the  symbol  "  =  "  here  means  "  determines." 


268    THE  TEACHING  OF   ELEMENTARY  MATHEMATICS 

"  5.  If  two  points,  A,  By  of  a  straight  line  a  lie  in 
a  plane  a,  then  every  point  of  a  lies  in  a. 

"  6.  If  two  planes,  a,  /3,  have  a  point  A  in  common, 
they  have  at  least  one  other  point  B  in  common. 

"7.  In  every  straight  line  there  are  at  least  two 
points,  in  every  plane  at  least  three  non-collinear 
points,  and  in  space  at  least  four  non-coplanar  points. 

"II.  Axioms  of  arrangement  (Anordnung),  denning 
the  concept  '  between.' 

"  i.  If  A,  B,  C  are  three  collinear  points,  and  B  lies 
between  A  and  C,  then  B  also  lies  between  C  and  A. 

"  2.  If  A  and  C  are  two  collinear  points,  there  is  at 
least  one  point  B  between  them,  and  at  least  one  point 
D  such  that  C  lies  between  A  and  D. 

"3.  Of  any  three  collinear  points,  there  is  one  which 
lies  uniquely  between  the  other  two. 

"  4.  Any  four  collinear  points,  A,  B,  C,  D,  can  be  so 
definitely  arranged  that  B  lies  between  A  and  C  and 
also  between  A  and  D,  and  that  C  lies  between  A  and 
D  and  also  between  B  and  D. 

"  5.  Suppose  A,  B,  C  to  be  three  non-collinear  points, 
and  a  a  straight  line  in  the  plane  ABC,  but  not  con- 
taining A,  By  or  C ;  if  then,  the  straight  line  a  passes 
through  a  point  within  the  line-segment  AB,  it  must 
also  pass  through  a  point  within  the  line-segment 
BC  or  through  a  point  within  the  line-segment  AC.1 

1  These   five   axioms  of  Group   II   were   first   investigated  by  Pasch 
(Vorlesungen  iiber  neuere  Geometrie,  Leipzig,   1882),  and  the  fifth  is 
.  especially  due  to  him. 


THE  BASES  OF  GEOMETRY  269 

"III.  Axiom  of  parallelism,  the  denial  of  which 
leads  to  the  non-Euclidean  geometry. 

"  IV.    Axioms  of  congruence. 

"i.  If  A,  B  are  two  points  on  the  straight  line  a, 
and  A1  is  a  point  on  the  same  or  another  straight 
line  a',  it  is  possible  to  find  on  a  given  side  of  a1 
from  A'  one  unique  point  B1  such  that  the  line-seg- 
ment AB  (or  BA)  is  congruent  to  the  line-segment 
A'B'.  .  .  . 

"2.  If  a  line-segment  AB  is  congruent  to  both  A'B1 
and  A"B",  then  A'B'  is  also  congruent  to  A"B". 

"3.  Let  AB  and  BC  be  two  segments  of  a,  without 
common  points;  let  A'B'  and  B'C'  be  two  segments 
of  a',  also  without  common  points;  then  if  AB  is 
congruent  to  A'B',  and  BC  is  congruent  to  B'C',  it 
must  follow  that  AC  is  congruent  to  A'Cf." 

4.  This  is  the  axiom   for   angles   corresponding  to 
axiom  2  for  segments. 

5.  This  is  the  axiom  for  angles   corresponding  to 
axiom  3  for  segments. 

"6.  If  for  two  triangles,  ABC  and  A' B'C'  these 
congruences  exist  (using  '  =  '  for  congruence), 

AB=A'B',  AC=A'C,  angle  BAC=  angle  B'A'C, 
then  must  these  also  exist, 
angle  CBA  =  angle  C'B'A',  angle  ACB  =  angle  A'CB*. 

"V.  Axiom  of  continuity  (Stetigkeit)  —  the  axiom  of 
Archimedes. 


2/0    THE  TEACHING  OF  ELEMENTARY  MATHEMATICS 

"  Let  A1  be  any  point  on  a  between  any  given 
points  A  and  B ;  suppose  A2,  A&  A±,  •••  so  taken 
that  A  l  lies  between  A  and  A2,  A2  between  A1  and 
A3,  etc.,  and  also  such  that  the  segments  AAl9  A^A^ 
A2A3,  •-  are  equal;  then  must  there  be  in  the 
series  A2,  Az,  A^  •••  a  point  An  such  that  B  lies 
between  A  and  An.  —  The  denial  of  this  axiom  leads 
to  the  non-Archimedean  geometry." 

Hilbert  inserts  the  necessary  definitions  for  under- 
standing these  postulates  (axioms),  and  adds  numerous 
corollaries  showing  the  far-reaching  effect  of  the 
statements ;  but  this  is  not  the  place  to  enter  this 
interesting  field.  Whether  or  not  his  postulates  are 
sufficient,  it  is  evident  that  tacitly  or  openly  they 
are  assumed  in  our  elementary  rectilinear  geometry. 
Their  consideration  should  convince  the  teacher  that 
the  question  of  the  postulates  is  by  no  means  the 
simple  one  which  the  text-books  sometimes  make  us 
feel. 

Thus  geometry  is  exact,  not  because  its  postulates 

necessarily    agree    with    the    facts    of    the    external 

world;   that  is  not  of  so  much  moment.      It  is  exact 

>  because  it  postulates   definitely  at   the   outset  certain 

:  few  statements  concerning  figures  in  space,  and  then 

applies    logic    to   see   what   other   statements    can    be 

deduced  therefrom. 


CHAPTER  XII 
TYPICAL  PARTS  OF  GEOMETRY 

The  introduction  to  demonstrative  geometry  may  well 
be  made  independent  of  the  text-book,  unless  the  book 
offers  some  special  preparatory  work.  If  the  pupils 
have  not  a  reasonable  knowledge  of  geometric  draw-s 
ing,  a  few  days  may  profitably  be  devoted  to  this  sub- 
ject exclusively.  Professor  Minchin  has  this  to  say  of 
the  English  schools,  and  the  same  is  almost  as  true  of 
our  own :  "  So  far  as  I  am  able  to  learn  by  inquiry, 
Euclid  is  taught  in  all  our  schools  without  the  aid 
of  rule,  compasses,  protractor,  or  scale.  This  is  quite 
in  accordance  with  the  traditional  method — the  classi- 
cal method  which,  unfortunately,  so  greatly  domi- 
nates English  education  —  and  quite  conducive  to 
long-delayed  knowledge  of  the  subject. 

"  Now  the  use  of  the  above  simple  instruments  for 
all  beginners  in  geometry  is  the  first  change  that  I 
advocate,  whether  we  continue  to  teach  from  Euclid's 
book  or  from  one  proceeding  on  simpler  and  better 
lines.  Well-drawn  figures  possess  an  enormous  teach- 
ing power,  not  merely  in  geometry,  but  in  all  branches 
of  mathematics  and  mathematical  physics."  l 

1  The  Teaching  of  Geometry,  The  School  World,  Vol.  I,  p.  161  (1899). 

271 


2/2     THE  TEACHING  OF  ELEMENTARY   MATHEMATICS 

Before  undertaking  the  ordinary  text-book  demon- 
strations the  teacher  will  also  find  it  of  great  value  to 
offer  a  few  preliminary  theorems  which  pave  the 
way  for  the  usual  sequence  of  propositions,  giving  a 
notion  of  what  is  meant  by  a  logical  proof,  and  creat- 
ing a  habit  of  working  out  independent  demonstrations. 
The  following,  for  example,  might  be  given  in  this 
way:  (i)  All  right  angles  are  equal  (if  the  text-book 
postulates  the  demonstrable  fact  of  the  equality  of 
straight  angles);  (2)  At  a  point  in  a  given  line  not 
more  than  one  perpendicular  can  be  drawn  to  that 
line  in  the  same  plane  —  not  that  one  can  be  drawn, 
as  so  many  text-books  affirm  but  fail  to  prove ;  (3)  The 
complements  of  equal  angles  are  equal ;  the  proposi- 
tion concerning  vertical  angles,  and  several  others  of 
the  simpler  ones  selected  from  the  first  "book." 

After  a  little  work  of  this  kind  the  pupil  is  prepared 
to  understand  the  nature  of  a  logical  proof.  Indepen- 
dence will  begin  to  assert  itself,  confidence  in  his 
ability  to  handle  a  proposition  without  a  slavish  depen- 
dence upon  his  text-book,  while  mere  memorizing  will 
fail  to  secure  the  usual  foothold  at  the  start.  These 
two  points  may  now  be  impressed :  (i)  Every  statement 
in  a  proof  must  be  based  upon  a  postulate,  an  axiom,  a 
definition,  or  some  proposition  previously  considered ; 
(2)  No  statement  is  true  simply  because  it  appears 
from  the  figure  to  be  true.  With  this  preliminary 
treatment  of  a  dozen  or  more  simple  propositions,  and 


TYPICAL  PARTS  OF   GEOMETRY  273 

with  some  instruction  concerning  geometric  drawing,^ 
the  text-book  sequence  may  be  undertaken  with  much 
less   danger   of   discouragement,  of   slovenly   work,  of 
groping  in  the  dark,  and  of  mere  memorizing. 

Symbols  —  The  contest  between  the  opponents  of  all 
symbols  and  the  advocates  of  mathematical  shorthand 
in  geometry,  as  in  other  branches  of  the  science,  is 
about  over.  In  England  Todhunter's  Euclid  is  giving 
place  to  the  Harpur,  Hall  and  Stevens,  McKay,  Nixon, 
and  others  which  make  extensive  use  of  symbols, 
while  in  America  Chauvenet's  excellent  work  has  had 
to  give  place  to  less  scholarly  but  more  usable  text- 
books. 

In  general  one  is  practically  bound  by  the  symbols 
in  the  book  in  the  hands  of  the  class.  A  few  notes 
upon  the  subject  may,  however,  be  suggestive.  In 
the  first  place,  only  generally  recognized  mathematical 
symbols  should  have  place ;  in  a  world-subject  like 
mathematics,  provincialism  is  especially  to  be  con- 
demned. We  may  think  that  ||  would  be  a  better  sign 
of  equality  than  =,  but  the  world  does  not  think  so, 
and  we  have  no  right  to  set  up  a  new  sign  language. 
In  this  respect  it  is  unfortunate  that  some  of  our 
American  writers  should  continue  to  use  the  provin- 
cial symbol  for  equivalence  (=o=),  not  only  because  it 
is  difficult  to  make,  but  because  it  has  no  standing 
among  mathematicians.  Indeed,  the  distinction  be- 
tween equal  and  equivalent  is  so  nearly  obliterated 


274    THE  TEACHING  OF  ELEMENTARY  MATHEMATICS 

in  our  language  that  many  teachers  now  use  the 
more  exact  term  "congruent"  for  what  some  English 
writers  call  "identically  equal,"  even  though  the  text- 
book in  their  classes  has  the  word  "equal."  The 
symbol  for  congruence  (^),  a  combination  of  the 
symbols  for  similarity  (~,  an  S  laid  on  its  side,  from 
similis)  and  equality  (=),  is  so  full  of  meaning  and  is 
so  generally  recognized  by  the  mathematical  world 
that  its  more  complete  introduction  in  elementary  work 
is  desirable.  It  is  certainly  not  open  to  the  objection 
of  novelty,  for  it  dates  from  Leibnitz,  nor  of  the  provin- 
cialism and  want  of  significance  which  characterize  the 
American  symbol  for  equivalence. 

The  modern  symbols  for  limit  (=,  still  in  its  provin- 
cial stage),  identity  (=),  and  non-equality  (=£),  in  addi- 
tion to  the  ordinary  algebraic  signs,  are  also  convenient. 
There  is  also  much  advantage  in  following  the 
modern  method  of  reading  angles  and  lines,  and  of 
lettering  triangles.  Among  the  ancients,  when  angles 
were  always  considered  as  less  than 
1 80°,  it  was  a  matter  of  little  moment 
whether  one  should  read  the  angle 

here  illustrated  AOB  or  BOA.     But 

now  that  we  recognize  angles  of  any 
number  of  degrees,  as  when  we  turn  a  screw  through 
90°,  1 80°,  270°,  360°,  450°,  ••-,  it  becomes  necessary  to 
distinguish  the  two  conjugate  angles  in  the  figure.  The 


TYPICAL  PARTS  OF  GEOMETRY  275 

obtuse  angle  is,  therefore,  read  AOB,  and  the  reflex 
angle  BOA,  counter-clockwise.  Pupils  brought  up  to 
this  plan  from  the  beginning  have  a  great  advantage 
in  accuracy  when  they  come  to  speak  of  figures  which 
are  at  all  complicated.  The  counter-clockwise  reading 
of  positive  angles  and  the  clockwise  reading  of  nega- 
tive ones  is  also  very  helpful  in  the  generalization  of 
propositions  in  the  earlier  books. 

It  is  also  of  great  advantage  to  recognize,  before 
the  pupil  has  gone  too  far,  the  distinction  between  the 
line  segments  AB  and  BA.  Negative  magnitudes  can 
no  longer  be  kept  from  elementary  geometry,  say  what 
we  may  about  pure  form  and  the  non-algebraic  treat- 
ment of  the  subject.  Pupils  understand  the  negative 
magnitudes  of  algebra  —  then  why  not  apply  this 
knowledge  to  geometry,  thus  opening  fields  both  new 
and  interesting?  By  so  doing,  a  mutually  helpful 
correlation  is  established  between  algebra  and  geom- 
etry, a  correlation  always  recognized  in  the  more  ad- 
vanced portions  of  the  science. 

The  advantage  of  uniformity  in  lettering  triangles 
ABC,  XYZ,  •••,  in  counter-clockwise  order,  and  of 
lettering  the  sides  opposite  A,  B,  C,  respectively,  a,  b,  c 
(and  so  for  ;r,  y,  zt  etc.),  is  apparent  to  all  who  have 
accustomed  themselves  to  the  arrangement. 

Reciprocal  theorems  —  There  is  an  interesting  line  of 
propositions,  early  met  by  the  pupil,  in  which  one  theo- 


276    THE  TEACHING  OF   ELEMENTARY  MATHEMATICS 

rem  may  be  formed  from  another  by  simply  replacing 

the  words 

point  by  line, 

line  by  point, 

angles  of  a  triangle  by  (opposite)  sides  of  a  triangle, 
sides  of  a  triangle  by  (opposite)  angles  of  a  triangle. 

This  is  seen  in  the  following  propositions : 

If  two  triangles  have  If  two  triangles  have 
two  sides  and  the  included  two  angles  and  the  includ- 
angle  of  the  one  respec-  ed  side  of  the  one  respec- 
tively equal  to  two  sides  tively  equal  to  two  angles 
and  the  included  angle  of  and  the  included  side  of 
the  other,  the  triangles  are  the  other,  the  triangles  are 
congruent.  congruent. 

If  two  sides  of  a  triangle  If  two  angles  of  a  triangle 
are  equal,  the  angles  oppo-  are  equal,  the  sides  oppo- 
site those  sides  are  equal.  site  those  angles  are  equal. 

Of  course  the  teacher  may  pass  over  this  relation- 
ship, as  most  text-books  do,  without  comment.  But 
there  is  great  advantage  in  recognizing  the  parallelism 
early  in  the  course,  for  two  reasons :  (i)  It  adds  greatly 
to  the  pupil's  interest  to  see  this  symmetry  of  the  sub- 
ject, to  note  that  certain  propositions  have  a  dual; 
and  (2)  It  often  suggests  new  possible  theorems  for 
investigation  —  the  pupil  has  the  interest  of  discov- 
ery. This  is  seen  in  the  following  simple  exercise:  In 
a  triangle  ABC,  where  a.-=  b,  the  bisector  of  angle  C, 


TYPICAL  PARTS  OF  GEOMETRY  277 

produced  to  c,  bisects  side  c.  The  pupil  who  is  led  to 
discover  the  reciprocal  theorem,  and  to  investigate  its 
validity  (for  reciprocal  statements  are  not  always  true), 
has  opened  before  him  a  field  of  perpetual  interest,  a 
field  in  which  he  is  an  independent  worker. 

Converse  theorems  are  often  thought  uninteresting. 
Students  get  the  idea  that  the  converses  are  always 
true,  and  that  it  is  a  stupid  waste  of  time  to  prove  them. 
And  yet,  so  necessary  are  these  propositions  to  the 
logical  sequence  of  geometry,  that  they  have  an  impor- 
tant place.  In  arranging  to  present  the  subject  to  a 
class,  the  teacher  is  met  by  three  questions:  (i)  What 
are  converse  theorems  ?  (2)  Are  converses  always  true  ? 
(3)  How  are  converse  theorems  best  proved  ? 

Two  theorems  are  said  to  be  converse,  each  of  the 
other,  when  what  is  given  in  the  one  is  what  is  to  be 
proved  in  the  other,  and  vice  versa.  For  example,  "  In 
triangle  ABC,  if  a  =  b  then  angle  A  =  angle  B"  and, 
"  In  triangle  ABC,  if  angle  A  =  angle  B  then  a  =  £," 
are  converses,  and  each  is  true ;  but  if  the  second  one 
should  read,  "  In  triangle  ABC ',  if  all  the  angles  are  equal 
then  a  =  b"  the  two  would  not  be  converses,  although 
what  is  given  in  the  first  (a  =  b)  is  what  is  to  be  proved 
in  the  second,  for  the  vice  versa  element  is  wanting. 

The  class  should  be  made  aware  of  numerous  false 
converses,  that  the  necessity  for  proof  may  be  appreci- 
ated. For  example,  "All  right  angles  are  equal  angles," 
"  If  a  triangle  contains  a  right  angle  it  is  not  an  equi- 


THE  TEACHING  OP^  ELEMENTARY   MATHEMATICS 

lateral  triangle,"  "  If  two  numbers  are  prime  their 
product  is  composite,"  are  all  true  statements,  but  their 
converses  are  not. 

There  are  so  many  converses  to  be  proved  that  the 
teacher  will  find  it  advantageous,  both  as  to  time  and 
logic,  to  consider  the  Law  of  Converse  rather  early  in 
the  course.  At  the  expense  of  one  or  two  lessons 
given  to  the  understanding  of  the  law,  the  time  should 
be  spared,  since  it  will  be  amply  repaid  later.  The  law 
is  as  follows : 

Whenever  three  theorems  have  the  following  relations, 
their  converses  must  be  true: 

1.  If  it  has  been  proved  that  when  A>B,  then  X>  F,  and 

2.  If  it  has  been  proved  that  when  A=B,  then  X=  F,  and 

3.  If  it  has  been  proved  that  when  A<By  then  X<  F, 
then  the  converse  of  each  is  true.     For 

If  X>  Y,  then  A  can  neither  be  equal  to  nor  less 
than  B  without  violating  2  or  3;  /.  A>B,  which 
proves  the  converse  of  i. 

If  X—  F,  then  A  can  neither  be  greater  nor  less 
than  B  without  violating  i  or  3;  .*.  A=B,  which 
proves  the  converse  of  2. 

If  X<  F,  then  A  can  neither  be  greater  than  nor 
equal  to  B  without  violating  i  or  2;  .-.  A<B,  which 
proves  the  converse  of  3. 

This  law,  proved  once  for  all,  enables  us  to  prove 
such  of  the  converses  as  we  need  in  elementary  geom- 


TYPICAL  PARTS  OF  GEOMETRY  279 

etry  without  using  the  tedious  demonstration  of  Euclid 
with  every  case.  For  example,  as  soon  as  it  has  been 
proved  that,  in  triangle  ABC,  if  A  =  B  then  a  =  b,  and 
if  A>B  then  a>b  (which,  by  mere  change  of  letters 
in  the  figure,  also  proves  that  if  A  <  B  then  a  <  b), 
this  law  shows  that  the  three  converses  are  true. 

Should  any  teacher  feel  that  this  is  too  difficult  for 
beginners,  it  should  be  noticed  that  the  proof  is  iden- 
tical with  that  usually  given,  but  it  is  here  merely 
set  forth  for  subsequent  use,  and  is  given  a  name. 

Generalization  of  figures  —  Until  recently  elementary 
geometry  seemed  afraid  to  consider  a  reflex  angle,  or 
a  concave  polygon,  or  an  equilateral  triangle  as  a 
special  case  of  an  isosceles  triangle,  to  say  nothing 
of  a  cross  polygon,  or  a  cylinder  with  a  non-circular 
directrix,  or  a  negative  line-segment.  But  our  best 
recent  works  have  presented  these  and  other  modern 
ideas  in  such  a  simple  fashion  that  their  general  in- 
troduction cannot  long  be  delayed.  It  is  not  at  all 
a  matter  of  the  text-book;  it  lies  with  the  teacher  to 
make  much  or  little  of  it,  and  scarcely  any  feature 
of  the  work  adds  more  interest,  develops  more  orig- 
inality, or  better  paves  the  way  for  future  progress. 
Take  the  familiar  theorem  that  the  sum  of  the  interior 
angles  of  an  «-gon  equals  n  —  2  straight  angles, 
stated,  of  course,  in  various  ways  and  with  more  or 
less  circumlocution.  After  it  has  been  proved  for 
the  simple  convex  figure,  the  teacher  may  ask  if  if 


28O    THE  TEACHING  OF  ELEMENTARY   MATHEMATICS 

is  true  in  case  one  angle  becomes  reflex ;  he  may 
then  move  the  vertex  back  until  the  angle  becomes 
straight,  and  ask  the  same  question.  Students  have 
no  trouble  with  such  questions,  and  they  readily 
follow  a  teacher  to  the  consideration  of  the  cross 
polygon,  a  case  best  presented  by  moving  the  vertex 
of  a  marked  angle  through  one  of  the  opposite  sides. 
The  case  of  the  sum  of  the  exterior  angles  of  a 
polygon  is  also  a  valuable  one  for  beginners.  If  the 
student  will  letter  the  angles  for  the  ordinary  convex 
polygon,  and  keep  the  same  lettering  when  it  becomes 

concave  or  cross,  he  will  find  that  the  proof  is  the  same 

»•  *          i%      *' 

for   all   cases.      When'  the   angle  AOB,   for   example 

(always  read  \counter-clockwise),  becomes  BOA,  it  is  to 
be  considered  negativ^fbut  otherwise%th&  jypof » is  quite 
unchanged.  Ifiteed,  the  one  practically  unvarying) 
principle  to  be  gi$en  the  student  is  this :  Letter  the  sim- 
ple figure  properly,  keeping  the  same  letters  in  all  trans- 
formations, and  the  proof  will  be  the  same  for  all  cases. 
The  principle  is  well  illustrated  in  the  case  of  the 
square  on  the  side  opposite  an  obtuse  angle  of  a 
triangle.  It  equals  the  sum  of  the  squares  on  the 
other  sides  plus  twice  a  certain  rectangle.  As  the  angle 
becomes  less  obtuse  this  rectangle  becomes  smaller;  if 
the  angle  becomes  right,  this  rectangle  vanishes  and 
the  theorem  becomes  the  Pythagorean ;  if  the  angle 
becomes  acute,  a  certain  projection  becomes  negative, 
making  the  rectangle  negative,  and  instead  of  having 


s  v 

or 
TYPICAL  PARTS  OF  GEOMETRY         OF        28l 


//ttj  twice  a  certain  rectangle  we  have  minus  twice 
that  rectangle,  the  proposition  becoming  the  one  con- 
cerning the  square  on  the  side  opposite  an  acute  angle.1 

This  generalization  of  typical  figures  materially 
lessens  the  detail  of  geometry.  For  example,  the 
propositions  concerning  the  measure  of  an  inscribed 
angle,  an  angle  formed  by  a  tangent  and  a  chord,  an 
angle  formed  by  two  chords,  or  two  secants,  or  a  secant 
and  a  tangent,  or  two  tangents,  are  all  special  cases 
of  a  single  theorem.  It  would  be  unwise  to  give  this 
general  theorem  first,  but  after  considering  the  cases 
of  an  inscribed  angle,  and  the  angle  formed  by  a  chord 
and  tangent,  classes  have  no  trouble  in  taking  the  gen- 
eral case  and  in  so  transforming  the  figure  as  easily  to 
get  the  special  cases  from  it.  The  proof  has  only  a 
couple  of  steps  in  the  most  general  form,  and  it  is  a 
waste  of  time  to  make  special  theorems  for  each  of  the 
various  simple  cases. 

The  proposition  concerning  the  "product"  of  the 
segments  of  two  intersecting  chords,  or  secants,  is  also 
one  which  is  often  extended  through  three  or  four 
theorems.  It  requires  only  two  steps  to  prove  the 
general  case.  If  a  pencil  of  lines  cuts  a  circumference, 
the  rectangle  (product)  of  the  two  segments  from  the 

1  Upon  this  set  of  theorems,  however,  the  teacher  should  read  the 
report  of  the  sub-committee  on  mathematics  in  the  Report  of  the  Com- 
mittee of  Ten,  Bulletin  No.  205  of  the  U.  S.  Bureau  of  Education,  p.  113. 
The  position  there  taken  is,  however,  open  to  very  serious  question. 


282    THE  TEACHING  OF  ELEMENTARY  MATHEMATICS 

vertex  is  constant  whichever  line  is  taken.  From  this 
theorem  four  or  five  others  come  as  special  cases  by 
simply  transforming  the  figure  slightly.  The  time  has 
surely  passed  for  fearing  so  valuable  a  phrase  as  "  pencil 
of  lines." 

These  few  illustrations  suffice  to  show  that  elemen- 
tary geometry  offers  a  field,  interesting  to  teachers  and 
pupils  alike,  for  simple  generalizations.  The  danger 
lies  on  the  one  side  in  always  attempting  to  give  the 
general  before  the  particular  (a  fatal  error),  and  on  the 
other  in  cutting  out  all  of  the  interest  which  comes 
from  generalization,  thus  falling  into  the  old  humdrum 
of  multiplying  theorems  to  fit  all  special  cases. 

Loci  of  points  —  Most  of  our  elementary  works  devote 
some  space  to  the  treatment  of  a  few  simple  loci  of 
points,  the  reciprocal  subject  of  "sets  of  lines"  being 
generally  regarded  as  hardly  worth  considering  at  this 
stage  of  the  student's  progress.  The  subject  is  of 
little  or  of  great  value,  depending  on  the  use  subse- 
quently made  of  it.  A  few  of  our  recent  text-books 
have  carefully  explained  the  term  "locus,"  and  have 
given  satisfactory  proofs  of  the  theorems,  but  the 
majority  fail  in  two  particulars,  and  as  to  these  a  few 
words  may  be  of  value. 

To  say  that  the  locus  of  points  (in  a  plane)  is  the 
line  containing  those  points,  is  entirely  inadequate, 
for  this  line  may  contain  other  points,  or  the  locus 
may  consist  of  two  or  more  lines,  or  of  a  line  and  a 


TYPICAL  PARTS  OF  GEOMETRY  283 

point  (as  in  the  locus  of  a  point  r  distant  from  a 
circumference).  Perhaps  the  best  plan  is  to  fall  back 
on  the  etymology  of  locus  (Lat.  place)  and  say,  The 
place  of  all  points  satisfying  a  given  condition  is 
called  the  locus  of  points  satisfying  that  condition  — 
giving  further  explanation  by  means  of  illustration. 

But  the  most  serious  error  usually  found  is  in  the 
proof.  "  In  proving  a  theorem  concerning  the  locus  of 
points  it  is  necessary  and  sufficient  to  prove  two  things  : 
(i)  That  any  point  on  the  supposed  locus  satisfies  the 
condition ;  (2)  That  any  point  not  on  the  supposed  locus 
does  not  satisfy  the  condition.  For  if  only  the  first 
point  were  proved,  there  might  be  some  other  line  in 
the  locus ;  and  if  only  the  second  were  proved,  the  sup- 
posed locus  might  not  be  the  correct  one."  A  text-book 
which  fails  in  these  points  should  be  discarded. 

Methods  of  attack  —  There  is  a  certain  value  in 
turning  a  pupil  into  a  chemical  laboratory,  after  he 
has  seen  some  experiments  performed,  and  there 
telling  him  to  discover  something  new,  or  to  find  the 
atomic  weight  of  some  substance.  He  will  fail,  but 
the  attempt  may  serve  to  broaden  his  ideas.  It  is 
also  of  some  value  to  hand  him  a  few  crystals,  tell- 
ing him  to  prove  that  they  are  this  or  that  kind  of 
salt,  leaving  him  to  his  own  devices.  But  the  teacher 
who  would  do  this  with  elementary  students,  who 
would  offer  no  general  directions  as  to  methods  of 
attack,  who  would  allow  a  student  to  wander  aim- 


284    THE  TEACHING  OF  ELEMENTARY   MATHEMATICS 

lessly  about,  groping  blindly  and  wasting  his  energies 
in  futile  attempts,  would  be  looked  upon  as  a  failure. 
And  yet  this  is  about  what  we  usually  find  in  a  class 
in  geometry ;  students  are  turned  loose  among  a  mass 
of  exercises,  and  are  told  to  invent  new  proofs,  to 
find  new  theorems,  to  solve  problems  and  prove  theo- 
rems entirely  new  to  them.  Their  only  hint  is  that  given 
by  the  demonstration  of  some  recent  proposition ;  their 
only  hope,  to  stumble  upon  the  proof  —  to  draw  the 
prize  ticket  in  the  lottery  without  too  great  delay. 

Mathematicians  do  not  proceed  in  any  such  way; 
they  call  to  their  assistance  all  the  general  methods 
possible,  and  to  the  teacher  of  geometry  this  should 
be  a  lesson.  The  discovery  of  theorems,  new  at 
least  to  the  pupil  and  probably  to  the  teacher,  is  an 
interesting  application  of  the  law  of  reciprocity 
already  mentioned.  Thus  if  a  student  knows  Pascal's 
"  mystic  hexagram "  (If  the  opposite  sides  of  an  in- 
scribed hexagon  intersect,  they  determine  three  col- 
linear  points),  it  is  but  a  step  to  rediscover,  in  the 
same  way  that  it  was  originally  found,  Brianchon's 
well-known  theorem.1 

1  The  teacher  will  find  this  theory  worked  out  fully  in  Henrici  and 
Treutlein's  Lehrbuch  der  Elementar- Geometric,  Leipzig,  1881,  3.  Aufl., 
1897,  — one  of  the  most  suggestive  works  on  the  subject.  An  excellent 
little  handbook  which  deserves  a  place  in  the  library  of  every  teacher 
of  elementary  mathematics  is  Henrici's  Elementary  Geometry,  Congruent 
Figures,  London,  1879, —  a  work  in  which  the  reciprocity  idea  is  brought 
out  quite  fully. 


TYPICAL  PARTS  OF  GEOMETRY  285 

But  it  is  to  methods  of  attack  in  the  treatment  of 
exercises  that  it  is  desired  to  direct  especial  attention. 
This  subject  has  received  much  consideration  at  the 
hands  of  Petersen,1  Rouche"  and  De  Comberousse,2  and 
Hadamard,8  and  the  following  suggestions  are  largely 
from  their  works.4 

1.  In  attacking  a   theorem   take   the  most   general 
figure   possible.     E.g.,   if  a   theorem   relates  to  a  tri- 
angle,  draw   a   scalene   triangle;    one  which   is   equi- 
lateral or  isosceles   often  deceives  the  eye  and  leads 
away  from  the  demonstration. 

2.  Draw  all  figures  as  accurately  as  possible.     An 
accurate  figure  often  suggests  a  demonstration.      On 
the  other  hand,  the  student  who  relies  too  much  upon 
the  accuracy  of  the   figure  in    the    demonstration  is 
liable  to  be  deceived. 

3.  Be   sure   that  what  is   given   and  what  is  to  be 
proved  are  clearly  stated  with  reference  to  the  letters 
of  the  figure.     Neglect  in   this   respect  is  a  fruitful 
cause  of  failure. 

4.  Then  begin  by  assuming  the  theorem  true;  see 
what  follows  from  that  assumption;    then  see  if  this 

1  Methods  and  Theories  of  Elementary  Geometry,  London  and  Copen- 
hagen, 1879. 

2  Traite  de  Geometric,  6  ed.,  Paris,  1891. 

8  Lemons  de  Geometric  elementaire,  Paris,  1898. 

4  The  immediate  source  is,  however,  Beman  and  Smith's  New  Plane  and 
Solid  Geometry,  Boston,  1899,  p.  35,  152,  to  which  reference  is  made  for 
further  details. 


286    THE  TEACHING  OF  ELEMENTARY   MATHEMATICS 

can  be  proved  true  without  the  assumption ;  if  so,  try 
to  reverse  the  process. 

5.  Or   begin   by  assuming   the   theorem   false,    and 
endeavor   to    show   the    absurdity   of    the    assumption 
(reductio  ad  absurdum). 

6.  To  secure  a  clearer  understanding  of  the  propo- 
sition to  be  proved  it  is  often  well  to  follow  Pascal's 
advice,    and    "substitute    the    definition    in    place    of 
the  name  of  the  thing  defined." 

7.  In   attempting    the    solution    of    a    problem    the 
method  of  analysis  suggested  in  4,  above,  will  usually 
lead   to   success.      Assume   the   problem   solved,   con- 
sider what  results  follow,  and  continue  to  trace  these 
results   until   a   known    proposition   is   reached ;    then 
seek  to  reverse  the  process. 

8.  One  of   the   most  fruitful  methods  of   attacking 
problems  is  by  means  of  the  intersection  of  loci.     So 
long  as  it  is  known  merely  that  a  point  is  on  one  line, 
its  position  is  not  definitely  determined;    but  if  it  is 
known  that  the  point  is  also  on  another  line,  its  posi- 
tion  may  (and   if    both    lines    are    straight   must)  be 
uniquely   determined.      For   example,   if  it  is   known 
that  a  point  is  on  a  certain  straight  line  and  a  certain 
circumference,  it  may  be  either  of  the  two  points  of 
intersection.     Thus,  in  a  plane,  to  find  a  point  equally 
distant  from  two  fixed  points,  A,  B,  and  also  equally 
distant  from   two   fixed    intersecting   lines,  x,  y\    the 
locus  of  points  equidistant  from  A  and  B  is  the  per- 


TYPICAL  PARTS  OF  GEOMETRY  287 

pendicular  bisector  of  AB\  the  locus  of  points  equi- 
distant from  x  and  y  is  the  pair  of  lines  bisecting  the 
angles  xy  and  yx\  since,  in  general,  the  first  line 
will  cut  the  other  two  in  two  points,  both  of  these 
points  answer  the  conditions. 

Petersen  gives  numerous  other  methods,  but  the 
above  suggestions  answer  very  well  for  all  cases  the 
student  will  meet  in  elementary  geometry. 

Ratio  and  proportion  —  In  the  treatment  of  this 
chapter  we  have  two  extremes  of  method.  First 
there  is  the  Euclidean,  purely  geometric,  scientific 
and  logical  to  the  extreme.  It  is  because  of  this 
treatment  that  English  teachers  sometimes  argue  the 
more  strongly  for  Euclid  —  although  in  practice  they 
never  use  the  chapter!  The  fact  is,  it  is  altogether 
too  difficult  for  beginners,  even  as  modified  by  the 
syllabus  of  the  Association  for  the  Improvement  of 
Geometrical  Teaching.  One  has  but  to  read  the 
Euclidean  definition  of  equal  ratios  to  be  assured  of 
this  fact:  "The  first  of  four  magnitudes  is  said  to 
have  the  same  ratio  to  the  second,  which  the  third 
has  to  the  fourth,  when  any  equimultiples  whatsoever 
of  the  first  and  third  being  taken,  and  any  equi- 
multiples whatsoever  of  the  second  and  fourth;  if  the 
multiple  of  the  first  be  less  than  that  of  the  second, 
the  multiple  of  the  third  is  also  less  than  that  of  the 
fourth:  or,  if  the  multiple  of  the  first  be  equal  to 
that  of  the  second,  the  multiple  of  the  third  is  also 


288    THE  TEACHING  OF  ELEMENTARY  MATHEMATICS 

equal  to  that  of  the  fourth ;  or,  if  the  multiple  of 
the  first  be  greater  than  that  of  the  second,  the 
multiple  of  the  third  is  also  greater  than  that  of  the 
fourth."1 

The  other  extreme  is  the  purely  algebraic  plan,  the 
one  adopted  by  most  American  text-book  writers,  a 
plan  entirely  non-geometric,  unscientific,  a  break  in 
the  logic  of  geometry,  but  so  easy  that  neither  teacher 
nor  pupil  need  do  much  serious  thinking  to  master  it. 
Occasionally  a  writer  inserts  a  proposition  at  the  end 
of  the  chapter,  intending  to  bridge  the  chasm  between 
algebra  and  geometry,  but  it  rarely  creates  any  im- 
pression upon  the  student. 

Between  these  extremes,  the  strictly  scientific  and 
the  strictly  unscientific,  the  too  difficult  and  the  too 
easy,  the  geometric  and  the  algebraic,  the  serious 
and  the  trivial,  there  is  at  least  one  fairly  scientific 
and  usable  mean.  It  consists  in  proving  that  there 
is  a  one-to-one  correspondence  between  algebra  and 
geometry,  with  this  relationship : 

Geometry.  Algebra. 

A  line-segment.  A  number. 

The  rectangle  of  two  line-  The  product  of  two  numbers, 
segments. 

This  having  been  made  a  matter  of  proof,  it  is 
further  postulated  that  any  geometric  magnitude  may 

1  Blakelock's  Simson's  Euclid,  London,  1856. 


TYPICAL  PARTS  OF  GEOMETRY  289 

be  represented  by  a  number.  With  these  assump- 
tions and  prbofs,  the  laws  of  proportion  may  be 
proved  either  by  algebra  or  by  geometry,  as  may 
be  the  most  convenient.  The  first  proposition,  stated 
in  dual  form,  would  then  read: 

If  four  numbers  are  in  If  four  lines  are  in  pro- 
proportion,  the  product  of  portion,  the  rectangle  of 
the  means  equals  the/;W-  the  means  equals  the  rec- 
^tct  of  the  extremes.  tangle  of  the  extremes. 

The  impossible  in  geometry  —  While  it  does  not 
enter  into  the  province  of  the  teacher  to  require  the 
pupil  to  attempt  the  impossible,  at  the  same  time 
the  questions  of  the  limits  of  the  possible  frequently 
arise  even  in  plane  geometry. 

To  say  that  nothing  is  impossible,  is  to  make  a 
pleasant  sounding  epigram,  and  if  it  means  that  it  is 
possible,  given  infinite  power,  to  do  any  particular 
thing,  it  is  true.  It  merely  asserts  that  nothing  is 
impossible  if  one  has  the  means  to  insure  its  possi- 
bility. But  the  moment  that  limitations  are  imposed, 
the  epigram  ceases  to  be  true.  To  draw  a  circle 
with  the  compasses  is  possible;  with  the  straight- 
edge only,  it  is  impossible.  To  draw  a  straight  line  is 
possible,  but  if  one  is  limited  to  the  use  of  the  com- 
passes it  becomes  impossible.  To  draw  an  ellipse, 
hyperbola,  cissoid,  or  conchoid,  —  all  these  are  pos- 
sible if  the  necessary  instruments  are  allowed,  but 
u 


2QO     THE  TEACHING  OF   ELEMENTARY   MATHEMATICS 

they  are  impossible  with  simply  the  compasses  and 
straight-edge. 

From  remote  antiquity  men  have  tried  to  trisect 
an  angle,  a  problem  simple  enough  if  the  necessary 
instruments  are  allowed,  but  one  well  known  by 
mathematicians  to  have  been  proved  to  be  impossible 
by  the  use  of  compasses  and  straight-edge  alone. 
It  is  not  that  the  world  has  not  yet  solved  it,  be- 
cause, like  the  fact  that  xn  +  yn  cannot  equal  zn  for 
n  >  2,  it  might  sometimes  yield  to  proof ;  but  it  has 
already  been  proved  that  it  cannot  be  solved.1 

Similarly  the  problem  of  constructing  a  square 
equal  to  a  given  circle,  "squaring  the  circle,"  is  easy 
enough  if  one  may  use  a  certain  curve,  but  it  has 
been  proved  to  be  impossible  by  the  use  of  the 
instruments  of  elementary  geometry.  In  the  same 
category  belong  the  problems  of  the  duplication  of 
the  cube,  and  the  construction  of  the  regular  hepta- 
gon. The  world  is  full  of  circle-squarers,  and  cube- 
duplicators,  and  angle-trisectors,  simply  because  these 
elementary  historic  facts  are  unknown. 

Solid  geometry  —  Euclid  paid  little  attention  to  solid 
geometry,  with  the  result  that  his  followers  in  the 
English  schools  have  also  neglected  it.  Since  the  con- 
servative Eastern  states  have  always  been  influenced  by 

1  Upon  this  and  other  problems  mentioned  in  this  connection,  the  most 
accessible  work  for  teachers  is  Klein's  Famous  Problems  of  Elementary 
Geometry,  English,  by  Beman  and  Smith,  Boston,  1896. 


TYPICAL  PARTS  OF  GEOMETRY  291 

the  educational  traditions  of  England,  solid  geometry  has 
never  had  the  hold  in  the  preparatory  schools  that  it 
has  in  the  Central  and  Western  states,  where  tradition 
counts  for  less.  The  argument  on  the  one  side  is  this : 
In  the  time  at  our  disposal  we  cannot  teach  all  of  plane 
geometry,  to  say  nothing  of  the  solid  —  as  if  all  of 
plane  geometry  could  ever  be  taught!  The  argument 
on  the  other  side  is  this :  The  whole  question  is  one  of 
degree ;  with  a  year  at  the  teacher's  disposal,  he  would 
do  better  to  teach  plane  geometry  about  two-thirds  of 
the  time,  and  solid  geometry  one-third ;  this  would  give 
mental  training  at  least  equally  valuable,  which  is  the 
first  consideration,  it  would  add  to  the  pupil's  interest, 
and  it  would  contribute  to  the  practical  side  through 
the  added  knowledge  of  mensuration. 

The  effort  has  several  times  been  made  to  work  out  a 
feasible  plan  for  carrying  solid  geometry  along  side  by 
side  with  the  plane.1  The  scheme  has  a  number  of 
advantages.  It  is  interesting,  for  example,  to  pass  a 
plane  through  certain  solids  (to  slice  into  them,  so  to 
speak),  and  get  figures  of  plane  geometry  out  of  them. 
It  is  also  interesting  to  note  the  one-to-one  correspond- 
ence between  the  spherical  triangle,  the  trihedral  angle, 
and  the  plane  triangle.  But  while,  theoretically,  this 
scheme  is  quite  feasible,  practically  it  has  few  followers. 
It  is  contrary  not  only  to  the  historical  development  of 
the  science,  but  also  to  psychology ;  it  makes  the  com- 

1  £&,  Paolis,  R.  de.  Element;  di  Geometria,  Torino,  1884. 


2Q2    THE  TEACHING  OF  ELEMENTARY   MATHEMATICS 

plex  contemporary  with  the  simple,  the  general  with  the 
particular,  from  the  very  first  It  is  interesting,  how- 
ever, to  see  how  skilfully  the  Italian  writers  are  han- 
dling the  matter. 

Practically,  it  has  been  found  best  to  take  up  the 
demonstrative  solid  geometry  after  a  course  in  plane 
geometry  has  been  completed.  The  subject  then  offers 
few  difficulties  to  most  students ;  a  little  patience  at  the 
outset,  a  few  simple  pasteboard  models,  easily  made  by 
the  class,  care  in  drawing  the  first  figures  so  as  to  bring 
out  the  perspective,  —  these  are  the  considerations  nec- 
essary in  beginning  work  in  the  geometry  of  three 
dimensions.  Models,  preferably  to  be  made  by  the 
student,  are  crutches  to  be  used  until  the  beginner  can 
walk,  then  to  be  discarded.  To  keep  them,  to  have 
special  ones  for  every  proposition,  even  to  have  their 
photographs,  is  to  take  away  one  of  the  very  things 
we  wish  to  cultivate, — the  imagination,  the  power  of 
imaging  the  solids,  the  power  of  abstraction.  In  gen- 
eral, the  appeal  to  models  should  be  made  only  as  it 
is  necessary  to  return  to  the  crutch  —  when  the  pupil 
falters. 

The  same  is  true  of  the  spherical  blackboard;  it  is 
valuable  and  should  be  used  in  every  school,  especially 
in  the  consideration  of  polar  and  symmetric  triangles ; 
but  never  to  depart  from  it  in  spherical  geometry,  or 
never  to  take  up  a  theorem  without  a  photograph  of 
the  sphere,  is  wholly  unwarranted  by  necessity  or  by 


TYPICAL  PARTS  OF  GEOMETRY  293 

the  demands  of  education.  The  student  needs  to  make 
abstractions,  to  get  along  with  a  figure  drawn  on  a  plane, 
and  to  be  able  to  work  independent  of  the  sphere  or 
its  photograph. 

The  teacher  will  do  well  to  add  to  the  treatment 
usually  given  some  little  discussion  of  recent  features  for 
which  we  are  indebted  to  the  Germans.  A  consider- 
able saving  is  effected  in  "producing"  lines,  planes,  and 
curved  surfaces,  in  treating  prisms,  pyramids,  cylinders, 
and  cones,  by  the  introduction  of  the  notion  of  pris- 
matic, pyramidal,  cylindrical,  and  conical  surfaces  and 
spaces.  The  concepts  are  simple,  and  by  their  use  a 
number  of  proofs  are  considerably  shortened.  The 
prismatoid  formula,  introduced  by  a  German,  E.  F. 
August,  in  1849,  should  also  have  place  on  account  of 
its  great  value  in  mensuration.  Euler's  theorem,  which 
states  that  in  the  case  of  a  convex  polyhedron  with 
e  edges,  v  vertices,  and  /  faces,  e  +  2  =/+  v,  also 
deserves  place,  both  for  the  reasoning  involved  and 
its  interesting  application  to  crystallography.  These 
additions  are  easily  made,  whatever  text-book  is  in 
use,  and  teachers  will  find  them  of  great  value.  The 
objection  on  the  score  of  difficulty  is  groundless. 

The  one-to-one  correspondence  between  the  poly- 
hedral angle  and  the  spherical  polygon  should  also  be 
noted,  a  correspondence  not  always  sufficiently  prom- 
inent in  our  text-books.  This  relation  may  be  set  forth 
as  follows: 


294    THE  TEACHING  OF   ELEMENTARY   MATHEMATICS 

"  Since  the  dihedral  angles  of  the  polyhedral  angles 
have  the  same  numerical  measures  as  the  angles  of  the 
spherical  polygons,  and  the  face  angles  of  the  former 
have  the  same  numerical  measure  as  the  sides  of  the 
latter,  it  is  evident  that  to  each  property  of  a  polyhedral 
angle  corresponds  a  reciprocal  property  of  a  spherical 
polygon,  and  vice  versa.  This  relation  appears  by 
making  the  following  substitutions: 

Polyhedral  Angles.  Spherical  Polygons. 

a.  Vertex.  a.  Centre  of  Sphere. 

b.  Edges.  b.  Vertices  of  Polygon. 

c.  Dihedral  Angles.  c.   Angles  of  Polygon. 

d.  Face  Angles.  d.  Sides. 

"  In  addition  to  the  correspondence  between  polyhe- 
dral angles  and  spherical  polygons,  it  will  be  observed 
that  a  relation  exists  between  a  straight  line  in  a  plane 
and  a  great-circle  arc  on  a  sphere.  Thus,  to  a  plane 
triangle  corresponds  a  spherical  triangle,  to  a  straight 
line  perpendicular  to  a  straight  line  corresponds  a  great- 
circle  arc  perpendicular  to  a  great-circle  arc,  etc."  It 
may  also  be  mentioned,  in  passing,  that  the  word  "  arc  " 
is  always  understood  to  mean  "great-circle  arc,"  in  the 
geometry  of  the  sphere,  unless  the  contrary  is  stated. 

A  further  relationship  of  interest  in  the  study  of 
solid  geometry  is  that  existing  between  the  circle 
and  the  sphere,  and  illustrated  in  the  following  state- 
ments : 


TYPICAL  PARTS  OF  GEOMETRY  295 

"The  Circle.  The  Sphere. 

A  portion  of  a  line  cut  off  by  A  portion  of  a  plane  cut  off  by 

a  circumference  is  a  chord.  a  spherical  surface  is  a  circle. 

The  greater  a  chard,  the  less  The  greater  a  circle,  the  less 

its  distance  from  the  centre.  its  distance  from  the  centre. 

A  diameter  (great  chord)  bi-  A  great  circle  bisects  the 

sects  the  circle  and  the  circum-  sphere  and  the  spherical  sur- 

fercnce.  face. 

Two  diameters  (great  chords)  Two  great  circles  bisect  each 

bisect  each  other.  other. 

Hence  may  be  anticipated  a  line  of  theorems  on  the  sphere, 
derived  from  those  on  the  circle,  by  making  the  following  substi- 
tutions : 

I.  Circle,  2.  circumference,  i.  Sphere,  2.  spherical  surface, 
3.  line,  4.  chord,  5.  diameter.  3.  plane,  4.  circle^  5. great  circle." 

The  advantage  in  noticing  this  one-to-one  correspond- 
ence is  evident  if  we  consider  some  of  the  theorems. 
In  the  following,  for  example,  a  single  proof  suffices 
for  two  propositions : 

If  a  trihedral  angle  has  If  a  spherical  triangle  has 

two  dihedral  angles  equal  two  angles  equal  to   each 

to  each  other,  the  opposite  other,    the   opposite   sides 

face  angles  are  equal.  are  equal. 

The  generalization  of  figures  already  mentioned  in 
speaking  of  plane  geometry  here  admits  of  even  more 
extended  use.  It  is  entirely  safe  to  take  up  the  men- 
suration of  the  volume  or  the  lateral  area  of  the  frus- 
tum of  a  right  pyramid,  and  then  let  the  upper  base 
shrink  to  zero,  thus  getting  the  case  of  the  pyramid 


296    THE  TEACHING  OF  ELEMENTARY  MATHEMATICS 

as  a  corollary,  or  let  it  increase  until  it  equals  the  lower 
base,  thus  getting  the  case  of  the  prism ;  the  prism 
would,  however,  naturally  precede  the  frustum.  So  for 
the  frustum  of  the  right  circular  cone,  and  the  cone  and 
cylinder,  a  method  not  only  valuable  from  the  consider- 
ation of  time,  but  also  for  the  idea  which  it  gives  of  the 
transformation  of  figures. 

Most  of  these  suggestions  can  be  used  to  advantage 
with  any  text-book.  Some  are  doubtless  used  already 
by  many  teachers,  and  it  is  hoped  all  may  be  of  value. 


CHAPTER  XIII 
THE  TEACHER'S  BOOK-SHELF 

Although  in  this  work  considerable  attention  has 
already  been  paid  to  the  bibliography  of  the  subject, 
a  few  suggestions  as  to  forming  the  nucleus  of  a 
library  upon  the  teaching  of  mathematics  may  be  of 
value.  It  has  been  the  author's  privilege,  after  lecturing 
before  various  educational  gatherings,  to  reply  to  many 
letters  asking  for  advice  in  this  matter,  and  so  he 
feels  that  there  are  many  among  the  younger  genera- 
tion of  teachers  who  will  welcome  a  few  suggestions 
in  this  line. 

In  the  first  place,  the  accumulation  of  a  large  num- 
ber of  elementary  text-books  is  of  little  value.  The 
inspiration  which  the  teacher  desires  is  not  to  be 
found  in  such  a  library ;  such  inspiration  comes  rather 
from  a  few  masterpieces.  Twenty  good  books  are 
worth  far  more  than  ten  times  that  number  of  ordi- 
nary text-books.  Hence,  in  general,  a  teacher  will 
do  well  never  to  buy  a  book  of  the  grade  which  he 
is  using  with  his  class ;  let  the  book  be  one  which 
shall  urge  him  forward,  not  one  which  shall  make 
him  satisfied  with  the  mediocre. 

297 


298     THE  TEACHING  OF  ELEMENTARY   MATHEMATICS 

Since  an  increasing  number  of  teachers,  especially 
in  our  high  schools,  have  some  knowledge  of  German 
or  French,  and  would  be  glad  to  make  some  use  of 
that  knowledge  if  encouraged  to  do  so,  it  should  be 
said  that  the  best  works  which  we  have  upon  general 
methods  of  attacking  the  various  branches  of  mathe- 
matics are  in  French.  The  best  works,  as  a  whole, 
illustrating  progress  in  particular  branches,  are  in 
German,  although  some  excellent  works  in  special 
lines  are  to  be  found  in  Italian.  The  other  Conti- 
nental languages  offer  but  little  of  value  that  has  not 
been  translated  into  English,  French,  or  German. 

Arithmetic  —  The  teacher  of  primary  arithmetic 
needs  to  consult  works  on  the  science  of  educa- 
tion rather  than  those  upon  the  subject  itself,  both 
because  all  of  our  special  writers  seem  to  hold  a  brief 
for  some  particular  device,  and  because  the  mathe- 
matical phase  of  the  question  is  exceedingly  limited. 
DeGarmo's  Essentials  of  Method  (Boston,  Heath)  and 
the  McMurrys'  General  Method  and  their  Method  of 
the  Recitation  (Bloomington,  Public  Sch.  Pub.  Co.)  are 
among  the  best  American  works.  Along  the  special 
line,  for  teachers  who  will  guard  against  going  to 
extremes,  may  be  recommended  Grube's  Leitfaden 
(translated  by  Levi  Seeley,  New  York,  Kellogg,  and 
by  F.  Louis  Soldan,  Chicago,  Interstate  Pub.  Co.), 
Hoose's  Pestalozzian  Arithmetic  (Syracuse,  Bardeen), 
Speer's  New  Arithmetic  (Boston,  Ginn),  and  Phillips's 


THE  TEACHER'S  BOOK-SHELF  299 

article  in  the  Pedagogical  Seminary  for  October,  1897. 
But  the  most  scholarly  work  upon  this  subject  that 
America  has  produced  is  McLellan  and  Dewey's  Psy- 
chology of  Number  (New  York,  Appleton),  a  work 
which  the  author  believes  to  go  somewhat  to  an  extreme 
in  its  ratio  idea,  but  one  which  every  teacher  should 
place  upon  his  shelves  and  frequently  consult. 

Along  higher  lines,  Brooks's  Philosophy  of  Arith- 
metic (Philadelphia,  Sower)  deserves  a  place.  Its  his- 
torical chapter  is  unreliable,  and  it  runs  too  much 
to  cases,  rules,  and  formulae,  but  it  has  many  good 
features,  and  it  is  worthy  of  recommendation.  As 
showing  the  views  of  recent  educators  as  to  what  mat- 
ter should  be  eliminated,  what  new  subjects  should 
be  added,  and  how  the  leading  topics  may  be  treated, 
the  author  ventures  to  suggest  Beman  and  Smith's 
Higher  Arithmetic  (Boston,  Ginn). 

In  French  there  is  little  of  value  upon  primary 
arithmetic.  Upon  higher  arithmetic,  however,  numer- 
ous works  have  appeared  which  cannot  fail  to  inspire 
the  teacher.  Of  these  the  best  is  Jules  Tannery's 
Lemons  d'Arithm&ique  th^orique  et  pratique  (Paris, 
Colin),  although  Humbert's  Traite"  d'Arithmetique 
(Paris,  Nony)  is  also  a  valuable  work.  For  one  who 
cares  to  go  into  the  theory  of  numbers  there  is  no 
better  introduction  than  Lucas's  Th£orie  des  Nom- 
bres  (Paris,  tome  i,  Gauthier-Villars). 

In  German  there  is  a  veritable  embarras  de  richesses. 


300    THE  TEACHING  OF  ELEMENTARY  MATHEMATICS 

The  number  of  works  upon  primary  arithmetic,  and  of 
text-books  designed  to  carry  out  particular  schemes,  is 
appallingly  great.  It  would  be  unwise  for  one  begin- 
ning a  library  to  attempt  to  purchase  this  class  of 
works.  It  is  better  to  put  upon  the  shelves  a  few 
works  which  weigh  these  various  methods,  presenting 
their  distinguishing  features  in  brief  compass.  The 
best  single  work  to  purchase  is  Unger's  Die  Methodik 
der  praktischen  Arithmetik  in  historischer  Entwickel- 
ung  (Leipzig,  Teubner),  the  latter  part  of  which  sets 
forth  the  characteristics  of  the  plans  suggested  by 
Pestalozzi,  Tillich,  Stephani,  Von  Turk,  Diesterweg, 
Grube,  Tanck,  Knillmg,  et  al.  A  second  work  of 
great  value  is  Janicke's  Geschichte  der  Methodik  des 
Rechenunterrichts,  which,  with  Schurig's  Geschichte 
der  Methode  in  der  Raumlehre,  forms  the  third 
volume  of  Kehr's  Geschichte  der  Methodik  des  Volks- 
schulunterrichtes  (Gotha,  Thienemann),  but  which  may 
be  purchased  separately.  A  third  workpfiardly  up  to 
those  mentioned,  however,  is  Sterner's  Geschichte  der 
Rechenkunst  (Miinchen,  Oldenbourg),  the  latter  part 
of  which  is  devoted  to  comparative  method.  For  the 
most  scholarly  treatment  of  arithmetic,  elementary 
algebra,  and  elementary  geometry,  as  of  other  sub- 
jects, by  grades,  the  teacher  should  own  a  copy  of 
Rein,  Pickel  and  Scheller's  Theorie  und  Praxis  des 
Volksschulunterrichts  nach  Herbartischen  Grundsatzen 
(Leipzig,  Bredt),  a  work  which  also  sets  forth  the 


THE  TEACHER'S  BOOK-SHELF  301 

German  bibliography  of  the  several  subjects.  Al- 
though advocating  a  particular  method,  and  therefore 
outside  of  the  general  province  of  this  bibliography, 
mention  should  be  made  of  Knilling's  latest  work, 
Die  naturgemasse  Methode  des  Rechenunterrichts 
in  der  deutschen  Volksschule  (Miinchen,  Olden- 
bourg),  on  account  of  its  psychological  review  of  the 
problem  of  elementary  arithmetic. 

Algebra  —  One  of  the  first  works  which  a  teacher 
may  profitably  own  is  Chrystal's  Algebra  (two  vol- 
umes, New  York,  Macmillan),  a  work  which  he  will  not 
soon  master,  but  a  fountain  from  which  he  will  get 
continual  inspiration.  Since  this  enters  but  little  into 
the  subject  of  the  equation,  it  should  be  supplemented 
by  Burnside  and  Panton's  Theory  of  Equations  (Dub- 
lin, Hodges).  To  these  may  well  be  added  that 
viultum  in  parvo,  Fine's  Number  System  of  Algebra 
(Boston,  Leach). 

The  most  scholarly  elementary  algebra  that  has 
appeared  in  recent  years  is  Bourlet's  Algebre  616men- 
taire  (Paris,  Colin),  a  work  which  is  thoroughly  up 
to  date  and  which  contains  a  large  amount  of  new 
matter  which  is  usable  in  high-school  work.  Of 
course  there  are  many  other  excellent  algebras  in 
French,  some  of  them  much  more  extensive  than 
Bourlet,  but  none  can  be  so  highly  recommended  as 
the  first  work  to  be  purchased. 

From   the   standpoint  of  method,  especially  as  ap- 


302     THE  TEACHING  OF  ELEMENTARY  MATHEMATICS 

plied  to  the  earlier  stages,  Schiiller's  Arithmetik  und 
Algebra  (Leipzig,  Teubner)  deserves  a  place.  It  is  a 
practical  book  by  a  practical  teacher.  German  works, 
however,  run  off  into  special  lines  to  such  an  extent 
that  it  becomes  difficult  to  select  a  small  number. 
For  the  teacher  who  is  taking  classes  through  literal 
equations,  and  who  wishes  to  somewhat  master  the 
subject,  Matthiessen's  Grundziige  der  antiken  und 
modernen  Algebra  der  litteralen  Gleichungen  (Leip- 
zig, Teubner)  will  prove  a  gold  mine,  but  it  is  not  at 
all  of  the  nature  of  a  text-book.  Quite  a  remarkable 
little  work,  condensing  the  modern  theory  of  equa- 
tions in  small  compass,  is  Petersen's  Theorie  der 
algebraischen  Gleichungen  (Kopenhagen,  Host).  If 
one  cares  to  look  into  higher  algebra,  Weber's  Lehr- 
buch  der  Algebra  (two  volumes,  Braunschweig ; 
Vol.  I,  French  by  Griess,  Paris,  Gauthier-Villars), 
or  Biermann's  Elemente  der  hohere  Mathematik 
(Leipzig,  Teubner),  are  the  best  of  the  recent  works. 
There  are  also  a  few  recent,  scholarly,  and  inexpen- 
sive works  published  in  the  Sammlung  Goschen  and 
the  Sammlung  Schubert  which  will  prove  of  value 
out  of  all  proportion  to  the  cost.  (See  p.  176,  note.) 
Geometry  —  The  teacher  of  geometry  should  have 
some  good  edition  of  Euclid.  On  account  of  its  second 
volume  on  solid  geometry  (Geometry  in  Space,  Oxford, 
Clarendon  Press),  Nixon's  may  be  recommended, 
although  the  Harpur  Euclid,  Hall  and  Stevens  (New 


THE  TEACHER'S   BOOK-SHELF  303 

York,  Macmillan),  and  others,  are  excellent.  As  an 
introduction  to  the  recent  development  of  elementary 
geometry,  Casey's  Sequel  to  Euclid  (Dublin,  Hodges) 
should  be  among  the  earliest  purchases,  and  to  this  may 
also  be  added,  with  profit,  three  recent  manuals  by 
M'Clelland  (Geometry  of  the  Circle,  Macmillan),  Du- 
puis  (Synthetic  Geometry,  Macmillan),  and  Henrici 
(Congruent  Figures,  London,  Longmans). 

In  France,  where  they  are  not  tied  to  Euclid,  nor 
even  to  Legendre,  there  is  more  flexibility  in  the  course 
than  is  found  in  England.  Accordingly  the  modern 
notions  of  geometry  have  more  readily  found  place,  and 
the  reader  of  French  will  find  some  very  inspiring 
literature  awaiting  him.  Probably  the  best  single  work 
for  the  teacher  of  geometry,  in  any  language,  is  Roucne* 
and  De  Comberousse's  Traite"  de  Ge'ome'trie  (Paris, 
Gauthier-Villars).  Of  the  recent  works,  Hadamard's 
Lemons  de  Ge'ome'trie  e'le'mentaire  (Paris,  Colin)  is  one 
of  the  best. 

In  Germany  still  more  flexibility  is  shown  than  in 
France.  The  making  of  geometry  an  exercise  in  logic, 
which  England  carries  to  an  extreme,  and  which  Amer- 
ica and  France  possibly  carry  too  far,  is  not  so  notice- 
able in  Germany.  The  result  is  a  shorter  course,  one 
divested  as  far  as  possible  of  propositions  in  the  nature 
of  lemmas,  but  one  in  which  modern  ideas  find  wel- 
come. To  appreciate  this  spirit  the  teacher  should 
purchase  Henrici  and  Treutlein's  Lehrbuch  der  Ele» 


304    THE  TEACHING  OF  ELEMENTARY   MATHEMATICS 

mentar-Geometrie  (Leipzig,  Teubner),  one  of  the  best 
books  published.  As  a  type  of  the  best  of  the  inex- 
pensive handbooks,  it  would  be  well  to  add  Mahler's 
Ebene  Geometrie  (Sammlung  Goschen,  Leipzig,  —  it 
costs  but  twenty  cents  in  Germany),  a  bit  of  concen- 
trated inspiration. 

Italy  has  produced  some  excellent  works  on  element- 
ary geometry;  indeed,  in  some  features,  it  has  been 
the  leader.  Socci  and  Tolomei's  Elementi  d'  Euclide 
(Firenze,  1899),  Lazzeri  and  Bassani's  Elementi  di 
Geometria  (Livorno,  1898),  Faifofer's  various  works 
(Venezia,  Tipog.  Emiliana),  and  Paolis's  Elementi  di 
Geometria  (Torino,  Loescher),  all  have  distinguishing 
features  which  would  entitle  them  to  a  place  upon  the 
shelves  of  the  reader  of  Italian. 

History  and  general  method  —  Probably  the  most 
practical  works  on  mathematical  history  to  purchase  at 
first  are  Ball's  (Macmillan)  and  Fink's  (Beman  and 
Smith's  translation,  Chicago,  Open  Court).  The  former 
is  the  more  popular,  the  latter  the  more  mathematical. 
Cajori  has  also  written  two  readable  works  upon  the 
general  subject  (Macmillan).  The  leading  works  are, 
however,  in  German,  and  have  been  mentioned  in  the 
foot-notes. 

On  general  method  the  pioneer  among  prominent 
writers  was  Duhamel,  whose  Des  Methodes  dans  les 
Sciences  de  Raisonnement  (Paris,  Gauthier-Villars)  fills 
five  volumes.  The  work  is  not,  however,  of  greatest 


THE  TEACHER'S  BOOK-SHELF  305 

practical  value  to  the  teacher  of  to-day.  Dauge's 
Cours  de  Methodologie  math£matique  (Paris,  Gauthier- 
Villars)  is  comparatively  recent,  but  this,  too,  fails  to 
touch  the  vital  points  in  which  the  elementary  teacher 
is  especially  interested.  Laisant's  La  Math^matique 
(Paris,  Carr£  et  Naud),  frequently  mentioned  in  this 
work,  is  a  small  volume,  but  it  is  one  of  the  best  efforts 
of  its  kind,  and  it  may  well  have  a  place  upon  the 
teacher's  book-shelf.  Clifford's  Common  Sense  of  the 
Exact  Sciences  (Appleton)  should  also  be  at  hand  for 
consultation. 

In  the  way  of  periodical  literature,  Enestrom's  Bib- 
liotheca  Mathematica  (Leipzig,  Teubner)  is  one  of  the 
best  publications  devoted  to  the  history  of  the  subject. 
As  to  general  mathematical  teaching,  Hoffmann's  Zeit- 
schrift  fur  mathematischen  und  naturwissenschaftlichen 
Unterrichts  (Leipzig,  Teubner),  and  L'Enseignement 
Math^matique,  Revue  Internationale  (bi-monthly,  Paris, 
Carr£  et  Naud),  are  among  the  best. 


INDEX 


[Of  several  foot-note  references  to  the  same  work,  only  the  first  is  given.] 


Aahmesu.    See  Ahmes. 
Abacus,  57,  101. 
yEneas  Sylvius,  13. 
Aggregation,  signs  of,  182. 
Ahmes,  xx,  54,  145. 
Alcuin,  16,  60,  61. 
Algebra 

in  arithmetic,  16,  17,  68,  124,  130. 

ethical  value  of,  169. 

growth  of,  145, 

kinds  of,  155. 

name,  151. 

practical  value,  168. 

what,  and  why  taught,  161,  165. 

when  studied,  170. 
Al-Khowarazmi,  151,  152,  201. 
Ailman,  228  «. 
Al-Mamun,  151. 
Al-Mansur,  150. 
Amusements  of  arithmetic,  15. 
Angle,  262,  274. 
Approximations,  142,  159. 
Arabic  numerals,  50,  52,  53. 
Arabs,  5,  151. 
Arbitrary  value  check,  190. 
Archimedes,  231,  238,  269. 
Argand,  213. 
Aristotle,  13,  47,  227. 
Arithmetic 

reasons  for  teaching,  i,  19,  79, 98. 

history  of  teaching,  71. 

when  to  begin,  116. 

utilities  of,  2,  7,  35. 

mediaeval,  58. 

crystallizing,  6/ti 


Arithmetic 

oral,  117. 

commercial,  7,  136. 

first  year  of,  114. 

applied  problems,  136. 

ancient  divisions,  56. 

present  status,  68. 

distinguished  from  algebra,  162. 
Arts,  seven  liberal,  4. 
Aryabhatta,  150. 
Ascham,  32. 
Assyrians,  5. 
Austrian  methods,  122. 
Axioms,  178,  257,  262. 

Babylonians,  5,  50,  225. 
Bachet  de  Meziriac,  15. 
Bagdad,  151. 
Bain,  24  n.,  28. 
Ball,  241 «.,  304. 
Beda,  7,  60. 
Beetz,  82  ft. 
Beman,  148*.  ,211*. 
Beman  and  Smith 

arithmetic,  66  ». 

algebra,  159  ». 

geometry.  285  n. 

trans,  of  Fink,  sow.,  304. 

trans,  of  Klein,  290*. 
Benedict,  St.,  60, 
Bertrand,  214. 
Bezout,  211. 
Biber,  80. 
Bibliography,  297. 
Biermann,  302. 


307 


308 


INDEX 


Blockmann,  80  ». 
Boethius,  10,  59. 
Bologna,  10. 
Bolyai,  265. 
Boncompagni,  53  n. 
Boniface,  St.,  60. 
Bourlet,  163  «.,  176,  219,  301. 
Brahmagupta,  200. 
Brautigam,  Bin. 
Bretschneider,  228  n. 
Brianchon,  232,  284. 
Brocard,  231. 
Brooks,  67  ».,  299. 
Browning,  12  «. 
Biirgi,  67  n. 

Burnside  and  Panton,  301. 
Business  arithmetic,  20. 
Busse,  58,  77. 

Cajori,  304. 
Calculi,  57. 

Calendar.    See  Easter,  61. 
Cantor,  G.,  106. 
Cantor,  M.,  nw. 
Capella,  59. 
Cardan,  14,  153. 
Carnot,  232. 
Cassiodorus,  59. 
Catalan,  41. 
Cauchy,  169. 
Charlemagne,  60. 
Chasles,  228  «.,  231,  232. 
Checks,  188. 
Chilperic,  59. 
Chinese,  2,  57. 
Chrystal,  163,  164  n.,  176,  189,  216,  301. 
Chuquet,  153. 

Church  schools,  5,  6,  15,  60,  62. 
Cicero,  6. 

Circle  squaring,  290. 
Clairaut,  240. 
Clarke,  6  n. 

Cloister.    See  Church  Schools. 
Colburn,  117. 
Comenius,  54. 

Committee  of  Ten,  69,  250,  281  n. 
Committee  of  Fifteen,  69,  70,  116. 
Compayre,  20  n.,  84  n. 
Complex  numbers.    See  Number  sys- 
tems. 


Compound  numbers,  22. 
Comte,  162,  186,  244. 
Conant,  44  «. 

Concentric  circle  plan,  88. 
Confucius,  33  n. 
Conrad,  14. 

Converse  theorems,  277. 
Correlation,  3. 

ounting,  45. 

ourt  schools,  59. 
Cube,  duplication  of,  290. 
Culture  value,  12,  20,  23,  27,  34, 39, 237, 

238. 
Cycloid,  area  of,  244. 

D'Alembert,  163,  220,  266. 

Date  line,  129. 

Dauge,  163  ».,  305. 

Davidson,  13  n, 

Decimals.    See  Fractions. 

Definitions,  28,  176,  257. 

De  Garmo,  no,  m».,  298. 

Degree,  177,  225. 

De  Guimp,  80. 

Delbos,  4«. 

De  Morgan,  44,  148  «.,  177  n.,  232. 

Denominate  numbers,  37,  65. 

Denzel,  88. 

Desargues,  231. 

Descartes,  231. 

De  Stael,  170. 

De  Tilly,  266  n. 

Dewey,  45».,  105,  299. 

Diesterweg,  18,  89. 

Diophantine  equations,  150. 

Diophantus,  148. 

Discount,  true,  35. 

Discovery,  method  of,  88. 

Dittes,  6». 

Division,  122. 

Dixon,  257  n. 

Dodgson,  229  n. 

Drawing,  241,  245,  271. 

Dressier,  120  n. 

Duhamel,  29^.,  304. 

Duplication  of  the  cube,  290. 

Dupuis,  303. 

Easter  problem,  5,  7, 62. 
Ebers,  10. 


INDEX 


309 


Egyptians,  10,  u,  12,  50,  145,  226. 
Elimination,  211. 

Encyklopadie  d.  math.  Wiss.,  29  n. 
Equation 

in  arithmetic,  16,  17,  68,  69, 124, 130. 

of  payments,  65. 

classification  of,  152. 

roots  of  numerical,  159. 

quadratic,  198. 

equivalent,  203. 

radical,  206. 

simultaneous,  208. 

diophantine.  150. 
Erfindungsmethode,  88. 
Euclid,  229,  235-238. 
Examinations,  10,  216. 
Exchange,  36, 65. 
Explanations,  140. 

Factor,  179. 
Factoring,  192, 197. 
Fahrmann,  46  «. 
Faifofer,  304, 
False  position,  124. 
Fermat,  41. 
Ferrari,  154. 
Ferro,  14,  154. 
Fibonacci,  53. 
Fine,  i86«.,  301. 
Fingers,  47,  58,  xox. 
Fink,  50*.,  304. 
Fiore,  14,  154. 
Fischer,  202. 
Fisher  and  Schwatt,  176. 
Fitch,  20«.,  24. 
Fitzga,  20  «. 
Formal  solutions,  123. 
Formal  steps,  in. 
.  Fractions,  n,  23,  54,  119. 

decimal,  55,  66, 119. 
Francke  Institute,  75. 
Frisius,  14,  ico». 
Functions,  162, 163, 

Galileo,  244. 
Galley  method,  67. 
Gaultier,  232. 
Gauss,  158,  213. 
Gemma  Frisius,  14,  icon. 
Generalization  of  figures,  279. 


Geometry 

history  of,  224. 

non-Euclidean,  233,  265,  369. 

defined,  234. 

limits,  236. 

why  studied,  237. 

in  the  grades,  239,  243. 

demonstrative,  250,  271. 

bases  of,  257. 

impossible  in,  289. 

solid,  290. 

inventional,  245. 
Gergonne,  232. 
Germain,  208. 
Gillespie,  162  n. 
Girard,  6  n. 
Girard,  Pere,  83. 
Goldbach,  41. 
Goodwin,  Bp.,  171. 
GOschen,  176,  302. 
Gow,  n».,  227. 
Graffenried,  65  «. 
Grammateus,  63. 
Graphs,  208. 
Grass,  97  n. 
Grassmann,  106. 
Greatest  common  divisor,  39. 
Greeks,  6, 12,  50,  51,  55,  150,  227. 
Greenwood,  125  «. 
Grube,  89,  118,  298. 
Grunert,  202. 
Guizot,  61  n. 

Hadamard,  285,  296,  303. 
Hall,  G.  S.,  140. 
Hall  and  Stevens,  302. 
Halliwell,  53». 
Hamilton,  95. 
Hankel,  106,  225  «. 
Hanseatic  League,  8,  62. 
Hanus,  139  ».,  244,  347. 
Harms,  92  n.,  245  n. 
Harpedonaptae,  226. 
Harpur  Euclid,  302. 
Harriot,  156  «. 
Harris,  125  n. 
Harun-al-Raschid,  15  L 
Hau  computation,  145. 
Heath,  148  «. 
Hebrews,  50. 


INDEX 


Heiberg,  263. 

Henrici,  O.,  189,  219,  237,  284  «.,  303. 

Henrici  and  Treutlein,  284  n.,  303. 

Henry,  53  «. 

Hentschel,  89,  113,  114  n. 

Heppel,  190  n. 

Herbart,  95,  in. 

Herodotus,  227. 

Heron,  148. 

Hilbert,  257  ».,  266. 

Hill,  19  «. 

Hindu  numerals,  50,  52,  53. 

History   of   mathematics,    I,  42,   145, 

224. 

Holzmuller,  173,  174,  251. 
Homogeneity  as  a  check,  191. 
Hoose,  8572.,  298. 
Homer,  160. 
Hoiiel,  229,  242. 
Hiibsch,  16. 

Hudson,  166,  167  ».,  170. 
Humanism,  13. 
Humbert,  299. 

Imaginaries.    See  Number  systems. 
India,  3. 
Induction,  244. 
Interest,  awakening,  220. 

simple  and  compound,  36. 
Interpretation  of  solutions,  220. 
Inventional  geometry,  245. 
Involution,  31. 
Isidore,  60. 

Janicke,  75,  8s«. 

Janicke  and  Schurig,  72  #. 

Jews,  5. 

Journal  Royal  Asiatic  Society,  52  w. 

Kallas,  92  «. 

Kant,  95,  265. 

Kaselitz,  92. 

Kawerau,  87. 

Kehr,  72**.,  300. 

Kepler,  66  «. 

Khayyam,  201. 

Khowarazmi,  151, 152,  201. 

Klein,  265,  290 n. 

Klotzsch,  ii4«. 

Knilling,  20 «.,  84,  86*.,  92,  94,  301. 


Kobel,  54. 

Konigsberger,  257  ». 
Konnecke,  53  n. 
Koreans,  57. 
Korner,  98  ». 
Kranckes,  88. 
Kriisi,  81. 

Laboratory  methods,  76. 

Lacroix,  240. 

Laisant,  n,  29;*.,  39,  49,  104,  140, 156, 

240,  305. 
Lange,  96. 
Langley,  245. 
Laplace,  235. 
Laurie,  3». 

Lazzeri  and  Bassani,  304. 
Lemoine,  231. 

Leonardo  Fibonacci  of  Pisa,  53. 
Liberal  arts,  seven,  4. 
Lobachevsky,  265. 
Loci,  282. 
Locke,  31. 
Lodge,  125  n. 
Logarithms,  67. 
Logic  in  mathematics,  24,  25,  167,  207, 

238,  239. 
Logistics,  56. 

Longitude  and  time,  34, 126. 
Loria,  229. 
Lucas,  299. 

Mac6,  32. 

Mahaffy,  13  «. 

Mahler,  304. 

Mamun,  151. 

Mansur,  150. 

Martin,  6». 

Mathews,  239  n. 

Matthiessen,  150  «.,  203,  302. 

McClelland,  303. 

McCormack,  176  «. 

McLellan  and  Dewey,  45  «.,  105,  299. 

McMurry,  no,  298. 

Mensuration,  137. 

Mental  gymnastic,  79,  84. 

Method,  rise  of,  74. 

great  question  of,  109. 

in  geometry,  283. 
Metric  system,  134. 


INDEX 


Meziriac,  15. 

Middle  ages,  58. 

Minchin,  251,  271. 

Minus  and  plus,  187. 

Mobius,  232. 

Mohammed  ben  Musa,  151,  152,  aoi. 

Mohammedans,  4. 

Muller,  45  ». 

Multiplication  and  division,  67,  74. 

Murhard,  1711. 

Napier,  67. 

Negative  numbers.    See  Number  sys- 
tems. 

Neuberg,  231. 
Newcomb,  260 *. 
Newton,  48. 
Nixon,  302. 

Non-Euclidean  geometry,  233. 
Notation,  48,  49,  112. 
Number  systems,  157,  184,  213. 

concept,  99. 

pictures,  77. 

Object  teaching,  71, 100, 102. 

Obsolete  in  arithmetic,  68,  69,  70. 

Odd  numbers,  57. 

Oliver,  Wait,  and  Jones,  176. 

Omar  Khayyam,  201. 

One-to-one  correspondence,  106,  113, 

288,  295. 

Oral  arithmetic,  117. 
Oriental  algebra,  150. 
Oughtred,  156  n. 
Oxford,  9. 

Paolis,  304. 

Paris,  University,  10, 

Paros,  9*. 

Partnership,  65. 

Pascal,  259  n. 

Payne's  translations,  20 «.,  84*. 

Perception,  78. 

Pestalozzi,  18,  48,  58,  78,  116. 

Petersen,  285,  302. 

Philanthropin,  76. 

Phillips,  93  n.,  298. 

T,  255,  256. 

Pincherle,  176,  219, 

Pitiscus,  66*. 


Pius  II,  13. 

Plato,  12,  227,  229, 235. 

Pliicker,  232. 

Plus  and  minus,  187. 

Poincare,  257*1. 

Poinsot,  163. 

Poncelet,  232. 

Postulates,  257,  262. 

Problems,  statement  of,  i8t 

applied,  in  algebra,  215. 
Problem  solvers,  14. 
Proklos,  233. 

Proportion.  36,  39, 129,  287. 
Puzzles,  40,  6x. 
Pythagoras,  13,  228. 

Quadratic  equations,  198. 
Quadrivium,  60. 

Rashdall,  5*. 

Ratio  idea  of  number,  48, 103. 

and  proportion,  129,  287. 
Reasons  for  teaching  mathematics,  x, 

12,    17,    20,    23,    27,  34,  39,  237, 

238. 

Rebiere,  126*. 
Rechenmeister,  9,  63. 
Rechenschule,  62. 
Reciprocal  theorems,  275. 
Recorde,  16,  156  ». 
Reidt,  32  ». 
Rein,  in,  247. 

Rein,  Pickel,  and  Scheller,  24*.,  300. 
Remainder  theorem,  195. 
Renaissance,  63. 
Reviews,  143. 
Rhyming  arithmetics,  73. 
Riese,  14. 
Rochow,  77. 

Roman  numerals,  50,  51,  54,  55. 
Rome,  6. 
Roots,  31. 

Rope  stretchers,  226. 
Rosen,  152  «. 
Rouche  and   De  Comberousse,  285, 

303- 

Rousseau,  240. 
Rudolff,  63. 
Riiefli,  86  n. 
Ruhsam,  1x8. 


312 


INDEX 


Rules,  31,  72,  130, 167. 

Russell,  257  *. 

Saccheri,  233. 
Safford,  124  ft. 
Sammlung  Goschen,  176. 
Schubert,  176. 
Scales  of  counting,  46. 
Schafer,  79. 
Schiller,  238. 
Schmid,  K.A.,  2». 
Schmidt,  K.,  6n. 
Schmidt,  Z.,  9. 
School  World,  239  ».,  252  n. 
Schotten,  260 n. 

Schubert,  176,  i86».,  265  ».,  302. 
Schuller,  302. 
Schurig,  300. 
Schuster,  124  «.,  245  n. 
Schwatt,  176,  239  n. 
Scratch  method,  67. 
Semites,  5. 
Servois,  232. 
Shaw,  245. 

Short  cuts,  137. 

Signs.    See  Symbols. 

Similar  figures,  261. 

Smith,  D.  E.,  50  «.,  66  «.,  158  «.,  159  «, 
285  ».,  290  #.,  304. 

Socci  and  Tolomei,  304. 

Socrates,  6. 

Solon,  12. 

Spartans,  6. 

Speer,  103  «.,  298. 

Spencer,  W.  G.,  245. 

Spencer,  H.,  27  «. 

Spiral  method,  118. 

Square  root,  31. 

St.  Benedict,  60. 

St.  Boniface,  60. 

Stackel  and  Engel,  264. 

Stammer,  20  «. 

Standard  time,  129. 

Staudt,  232. 

Stehn,  13,  14  n. 

Steiner,  232. 

Sterner,  6  «.,  300. 

Stevin,  66  n. 

Straight  line,  258. 
Sturm,  9. 


ubtraction,  xax. 

ully,  31  n. 

urd,  1 80. 

wan  pan,  57. 

ylvius,  13. 

ymbols,  66,  148,  155,  182,  222,  273. 

ymmetry  as  a  check,  191. 

Tacitus,  58,  59. 
Tanck,  92,  94. 

'annery,  162  n.,  299. 

'artaglia,  14,  154. 
Jeachers'  failures,  26. 
Text-books,  70,  139, 173,  254. 
Thales,  227,  228. 
Theon  of  Alexandria,  131 ». 
Tillich,  31,  77,  82,  86. 
Time,  34, 126. 

Tradition,  10. 
Trapp,  76. 

Trigonometry  in  algebra,  202. 
True  discount,  35. 
Turk,  87. 

Twelve  as  a  radix,  48. 
Tylor,  45  n. 

Unger,  7«.,  300. 

Universities,  8,  9. 

Utilities  of  arithmetic,  2,  20,  39. 

Veronese,  257  n. 
Vienna,  10. 
Vieta,  156,  201. 
Voltaire,  240. 
Von  Busse,  58,  77. 
Von  Rochow,  77. 
Von  Staudt,  232. 

Wagner,  63. 
Walker,  39  «.,  116. 
Wallis,  156  n. 
Ward,  42  «. 
Weber,  302. 
Weierstrass,  106. 
Wessel,  158,  213. 
Wordsworth,  51 ». 

Young,  174  n. 
Zahlenbilder,  77. 


A  HISTORY 

OF 

MATHEMATICS, 


BY 


FLORIAN  CAJORI,  Ph.D., 

Formerly  Professor  of  Applied  Mathematics  in  the  Tulaoe  University  ol 
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